THE NONCOMMUTATIVE CARATHEODORYFEJER PROBLEM
By
SRIRAM BALASUBRAMANIAN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2010
2010 Sriram Balasubramanian
I dedicate this dissertation to my family and friends.
ACKNOWLEDGMENTS
First and foremost, I would like to express my deepest gratitude to my advisor Dr.
Scott McCullough, for his everpresent guidance, encouragement and help, without
which this dissertation would never have been completed. I am forever grateful to him
for having me as his student and for inspiring me with his tremendous knowledge and
expertise in Mathematics.
I would also like to thank Dr. James Brooks, Dr. James Hobert, Dr. Michael Jury
and Dr. Lichien Shen for serving on my doctoral committee. My special thanks to Dr.
Michael Jury for his many valuable suggestions.
Finally, I would like to thank my family and my friends especially Balaji, Krishna,
Subbu and Vijay, for their constant encouragement and support.
TABLE OF CONTENTS
ACKNOWLEDGMENTS ................... ............... 4
ABSTRACT. ..................... .................... 7
CHAPTER
1 INTRO DUCTION ..................... .............. 8
1.1 Sum m ary of Results .. .. .. .. .. .. .. 14
1.2 Organization ................... ............... 16
2 MATRIX CONVEXITY ................... ............ 18
2.1 Matrix Convex Sets in Cd ......................... .. 18
2.1.1 Exam ples . .... 19
2.1.2 Properties .................... .. ............. 19
2.2 Matricial HahnBanach Separation ................... 21
2.3 The Noncommutative Fock Space and Creation Operators ... 26
2.3.1 The Free Semigroup on d Letters and Intial Segments ...... .26
2.3.2 The Noncommutative Fock Space. 27
2.3.3 The Creation Operators ..... ... 27
3 ABSTRACT OPERATOR ALGEBRAS ....................... 29
3.1 Abstract Operator Algebra ................ ......... 29
3.1.1 Examples .................. ............ 31
3.1.2 The Quotient Operator Algebra ..... ... ... 31
3.2 Representations of Abstract Unital Operator Algebras .... 34
4 THE ABSTRACT OPERATOR ALGEBRAS A(/C)" & A(C)"/I(/) ..... ..35
4.1 The Algebra A(KC)" of Scalar Formal Power Series .. 35
4.1.1 Formal Power Series ... 35
4.1.2 The Vector Space A(KC)" .............. ......... .. 36
4.1.3 Matrix Norms on A(/C)" ........................ 37
4.1.4 The Algebra A(KC) ........................... 39
4.2 Weak Compactness and A(KC)" ....................... 44
4.3 The Abstract Operator Algebra A(C) .. 47
4.4 Completely Contractive Representations of A(C) ... 48
4.5 The Abstract Operator Algebra A(C) / (C) .. 53
4.6 Attainment of Norms of Classes in Mq(A(1C)")/Mq(I(1C)) ... 54
5 THE NONCOMMUTATIVE CARATHEODORYFEJER PROBLEM ...... ..56
5.1 The CaratheodoryFej6r Interpolation Problem (CFP) .... 56
5.2 The Matrix Version .......... ........ ............. 56
5.3 The Operator Version ............... .......... .. 58
6 INFINITE INITIAL SEGMENTS ............... .......... 62
6.1 Examples of Noncommutative Operator Domains .... 62
6.2 The ddimensional Noncommutative Polydisc ... 64
6.3 The d x d Noncommutative Mixed Ball ... 69
7 FUTURE RESEARCH ................... ............. 72
R EFER EN C ES . . 73
BIOGRAPHICAL SKETCH ................... ............. 76
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
THE NONCOMMUTATIVE CARATHEODORYFEJER PROBLEM
By
Sriram Balasubramanian
August 2010
Chair: Scott McCullough
Major: Mathematics
We pose the CaratheodoryFej6r interpolation problem for open, circled and
bounded matrixconvex sets in Cd. By using the BlecherRuanSinclair characterization
of an abstract unital operator algebra, we obtain a necessary and sufficient condition for
the existence of a minimumnorm solution to the problem.
CHAPTER 1
INTRODUCTION
A classical interpolation problem in function theory is the CaratheodoryFej6r
interpolation problem (CFP): Given n + 1 complex numbers co, cl,..., c, does there exist
a complex valued analytic function f(z) = Yo fjzJ defined on the open unit disc D c C
such that j = cj for all 0
The problem and some of its variants were studied by Caratheodory, Fejer and
Schur during the early 20th century in [30], [31] and [8] A necessary and sufficient
condition for the solvability of the CFP, which is commonly referred to as the Schur
Criterion, is that the Toeplitz matrix
co 0 ... 0
(1.1)
0
Cn ... Ci Co
is a contraction.
An operator theoretic view of the CFP was first presented by Sarason in his
pioneering work [29]. His formulation has had a major impact not only on the CFP
and the related Pick interpolation problem but the development of operator theory and
the study of nonself adjoint operator algebras generally. We now present a proof of
the equivalence between the solvability of the CFP and the Schur Criterion that uses
Sarason's ideas. We will begin with some definitions and state some wellknown facts
(without proofs).
Let H2(D) denote the Hardy Hilbert Space defined by
H2(D) = f : D C : f(z) = az" and I a,,2 < co .
n=O nO=
The inner product on H2(D) is given by
00
(f, g) = anbn
n=0
where f(z) = anz" and g(z) = bnz.
n=O n=O
Let L2 denote the Hilbert space of squareintegrable functions on T with respect to
the normalized Lebesgue measure. The inner product is defined as:
1 027T
(f g) = 2 j f(e')g(e'o) dO
where dO denotes the Lebesgue measure on [0, 27].
As is often done, we will view L2 as a space of functions rather than as a space
of equivalence classes of functions, by identifying two functions to be equal if they are
equal a.e. with respect to the normalized Lebesgue measure. i.e.
L = f : T  C : I f(e0) 2d < oI .
The Hilbert space L2 is separable with orthonormal basis {en}nez, where en : T I C
is defined by en(ei') = ei'0. For a proof of this fact see [7]. The expansion of the function
f e L2 with respect to this orthonormal basis is called the Fourier Series expansion of f.
We write
f(e'0) > Y f(n)e'"
n=oo
where the Fourier coefficients f(n) are given by
1 2 (eifle ii
f(n) = (f, en) = OdO
and they satisfy
>I(n)12 < cc.
noo
The Fourier Transform produces a canonical unitary equivalence between H2(D)
and a subspace of L2 as described below. It is a consequence of Fatou's Theorem
(see pp 15 20, [22]) that for f c H2(D), the radial limit f(e'0) = lim f(re') exists
r l
for almost all 0. We will call f the boundary function of f. An important property of f is
that f c L2. The linear mapping that takes f c H2(D) to f e L2 is an isometry from
H2(D) onto a closed subspace of L2 which we will denote by H2. If g e L2 is given by
00
g(e'0) a g(n)e'"0, then it is well known that the function f defined on the unit disc
n=0
by f(z) = g(n)z" is in H2(D) and in addition, f = g. Conversely, if f e H2(D) is
n=0
given by f(z) = fz" then, the Fourier series expansion of f is f(e'") f ,e'"0.
n=O n=0
This equivalence gives us the following way to view the space H2 in terms of Fourier
coefficients, namely,
H2 = {g L2: g(n) = 0 for n < 0}.
Let f be a Lebesgue measurable function defined on 'T. The essential supremum of
f is defined by IIf = inf{M > 0 : f(z) < M a.e.}. Let L" denote the Banach space
L" = {f : T I C : fis measurable and lfll < oo}
and let H" denote H2 q L0.
Remark 1.0.1. If f is a bounded analytic function defined on the unit disc, then f e H"
and 1f  < sup{f(z) : z e D}. Conversely, iff c H", then the function f c H2(ID) is
bounded by I f  .
For ce LPc, the Toeplitz Operator with symbol 4 is defined by
Tof = P(Of)
for each f e H2, where P is the orthogonal projection of L2 onto H2. If ec H00, then
Tof = P(Of) = Of.
In this case the Toeplitz operator is said to be analytic and its matrix with respect to the
basis {ei 0} is given by
An important property of the Toeplitz operator To is that
11 Tll = 11 110
(1.2)
Lemma 1.0.1 (Sarason). Let
(i) U represent the Unilateral shift on H2, i.e. (Uf)(z) = zf(z) for all f c H2
(ii) K = H2 Q H2 where b is an inner function, i.e. b e H" and I (z) = 1 a.e.
(iii) S = PU K where P denotes the orthogonal projection of H2 onto K.
(iv) T be a (bounded) operator on K.
If TS = ST, then there exists a function ce H" such that T = PT K where To is the
analytic Toeplitz operator with symbol 0 and I1011, = II TII.
A proof of the above lemma can be found in [29].
Theorem 1.0.1. The CFP has a solution if and only if the matrix in equation (1.1) has
norm at most one.
Proof. Suppose that there exists a solution f to the CFP. It follows from remark 1.0.1
and equation (1.2) that f e H" and,
1 > sup{f(z)l :z C D} > f  = II T II.
Since the matrix in equation (1.1) is the compression of the analytic Toeplitz operator TT
to the subspace spanned by the orthonormal set (l, z, ..., z") of H2 (and computed
with respect to this basis), it follows that the norm of that matrix is at most 1.
For the converse, we apply Lemma 1.0.1 with b(z) = z"+1, K = (l, z, z ..., z")
and T : K K being the matrix in (1.1). Suppose that II TI < 1. Let S and P be the
operators that appear in the hypothesis of Lemma 1.0.1. The matrix of S with respect to
the basis {l, z, z ... z"} is given by
0
S = (1.3)
1 0
Thus TS = ST and Lemma 1.0.1 implies that there exists c e H" such that T = PTI K
and
11 11. = II Tll.
Since ce H", = f for some f c H2(D). It follows from Remark 1.0.1 that
sup{f(z)l : z eD} < 11 .11
Hence this f is a solution to the CFP. D
From the operator theory/algebra point of view, the CFP is essentially unchanged
if the coefficients co, c, ..., cn are taken to be elements of B(U) for some separable
Hilbert space U. Indeed, even in this case, a necessary and sufficient condition for the
solvability of the CFP is the same as before, only now the entries of the Toeplitz matrix in
(1.1) are bounded operators in B(U) and the matrix itself is an element of B(( n U). An
alternate way of viewing the Schur Criterion which is more convenient for our purposes
is the following.
Lemma 1.0.2. Let p(z) = YJno cjzJ e B() and di be an arbitrary Hilbert space. The
Schur Criterion is equivalent to the contractivity of the operator p( T) = Y:n cj 0 TJ
B(U H) for every contraction T E B(H) which is nilpotent of order n + 1; i.e., T"+ = 0.
Proof. (=) Let S be the matrix in equation (1.3). Since S is a contraction and S"+1 = 0
we have Ip(S)l < 1. The fact that the norm of the Toeplitz matrix in equation (1.1) is
equal to the norm of the operator p(S) completes the argument.
(=) Fix a Hilbert space H and a contraction T e B((() which satisfies T"+1 = 0.
Let DTr = (I TT*)2. DTr is called the defect operatorof T*. Let {e}j}jo denote the
standard orthonormal basis of C"+1. Define the operator V: H C"+1 '0 by
n
Vh= ej e DT.(T*yh.
j=o
Then
n n
(Vh, Vh)= ( ej DT*(T*yh, ek DT*(T*)kh)
j= 0 k= 0
n
= Z(DT.(T*yh, DT*(T*)Jh)
j=o
n
= {(TJ(/ TT*)(T*)Jh, h)
j=o
= (h, h).
Thus V is an isometry. Moreover, For each k = 0, 1,2,... we have,
n
V(T*)kh = Y e, DTr(T*y+kh
J=o
nk
= e, DTr(T*Y kh
j=o
n
= ((S*)k I)( eg j DT*(T*)Jh)
j=o
((S*)k l) Vh.
This implies that for each k = 0, 1, 2,...,
Tk =V*(Sk l)V. (1.4)
Using equation (1.4) we get,
n
p(T)= II c, : V*(S /H)V
j= 0
n
= I1(G/0 v*)(Y C 0 (Sj 0/H)) (G V)ll
=o0
j 0
= 1p(S)l
<1.
1.1 Summary of Results
Several commutative multivariable generalizations of the CFP have been obtained
for different domains the polydisc Dd c Cd for example and for different interpolating
classes of functions, for example the SchurAgler class of analytic functions that
take contractive operator values on any dtuple of commuting strict contractions in a
manner discussed in [1]. For more details see [14], [6]. Some results on the problem
for bounded circular domains in Cd can also be found in [11]. Some noncommutative
generalizations of the CFP have also been studied in [25], [26], [9], [20], [4].
In this thesis, some of the existing results on the CFP have been extended to
the noncommutative setting of the free algebra on a finite number of generators. An
example of a domain we consider here is the d x d noncommutative matrix mixed ball
defined by
Ddd = U{X = (X1, X2.... Xd) : Xi are n x n matrices and Xop < 1}.
nEN
where IXop is the norm of the operator X = (Xy)d d1 (Cn)ad (C)d
What follows are some definitions which will lead us to the statement of the CFP for
this particular domain.
* Let FdT denote the semigroup of words generated by the symbols {g,~})jd1
* A set A c TFd is said to be an initial segment if wg,, gw A for all w A,
1 < i < d, 1
For X e Ddd and w = gilg,... gij, c Fda, the evaluation of X at w is defined by
XW = X"l, X"j,2 ... Xik .
X e Ddd is said to be Anilpotent if Xw = 0 for all w A.
A formal power series f is an expression of the form f = C,, fww where the
coefficients fw are complex numbers
For X C Ddd we define f(X) = J H0 C wl= fwXW whenever the series converges
converges (in the operator norm) in the indicated order.
The CFP for the d x d noncommutative matrix mixed ball is the following: Let A, a
finite initial segment, and
P = Pww
wEA
be given. Does there exist a formal power series f that fw = pw for w e A and
sup{f(X) : X C d} < 1?
(A special case of) Our main result is the following.
Theorem 1.1.1. There exists a (minimumnorm) solution f to the above problem if and
only if
sup{p(X)  : X e Ddd, X is A nilpotent} < 1.
In the body of the thesis we actually pose the CFP for more general domains that
are matrix convex sets in Cd, and using the BlecherRuanSinclair characterization of
abstract operator algebras, prove a generalization of Theorem 1.1.1 which allows for
operatorvalued coefficients pw and fw.
Throughout this thesis, we will be working with noncommutative analytic functions,
i.e. formal power series with matrix or operator coefficients that converge on some
noncommutative neighborhood of the origin (see [32], [33], [34], [24], [25], [26], [21],
[19]). These functions are not only objects of great mathematical interest, but have
applications in areas such as control theory and optimization. Noncommutative
polynomials, in particular, are of special interest since noncommutative polynomial
inequalities (matrix inequalities where the unknowns are matrices too), occur naturally
in the context of dimensionfree linear systems. Recent advances in the study of
noncommutative linear matrix inequalities can be found in [12], [18].
1.2 Organization
This thesis is organized as follows: In Chapter 2, the definition of a matrix
convex set in Cd is given along with some examples and properties and a proof of
the EffrosWinkler matricial HahnBanach Separation Theorem. The chapter ends with a
discussion on the Noncommutative Fock Space and the associated Creation Operators.
In Chapter 3, the definition an abstract operator algebra is introduced along with
some examples. It is also shown that the quotient of an abstract operator algebra by
a closed twosided ideal is an abstract operator algebra. The chapter ends with the
statement of the BlecherRuanSinclair Theorem for abstract unital operator algebras.
Chapter 4 is where the interpolating class A(IC)" and the ideal Z(IC) are introduced.
It is shown that A(/C)" and the quotient A(/C)/IZ(/C) are abstract operator algebras.
Several key properties including a weakcompactness type property of the algebra
A(KC)" and the norm attainment property of the algebra A(/C)"/IZ(/) are established.
The chapter ends with a discussion on completely contractive representations of the
algebra A(KC)", where it is shown that tuples of finitedimensional compressions of
operators that give rise to completely contractive representations of A(KC)" lie on the
boundary of the underlying matrix convex set.
In Chapter 5, the matrix and the operator versions of the CFP (with a finite
initial segment A) for a class of matrixconvex sets in Cd are posed and using the
results from Chapters 2 4, a necessary and sufficient condition for the existence of a
minimumnorm solution is obtained.
In Chapter 6, a version of Theorem 1.1.1 where it is assumed that the initial
segment A is an infinite set, is proved for two special noncommutative domains namely,
the ddimensional noncommutative (operator) polydisc and the d x d noncommutative
(operator) mixed ball.
In Chapter 7, two important and very interesting questions that came up while this
work was in progress, are posed.
CHAPTER 2
MATRIX CONVEXITY
A basic object of study in this thesis is a quantized, or noncommutative, version of
a convex set. While the definitions easily extend to convex subsets of arbitrary vector
spaces, here the focus is on subsets of Cd, the complex ddimensional space. In this
chapter we present the definition of a matrix convex subset of Cd and introduce our
standard assumptions regarding these sets. Some properties and examples of such
sets and a matricial HahnBanach separation result are also presented. The chapter
ends with a discussion of the Creation Operators on the Noncommutative Fock Space.
2.1 Matrix Convex Sets in Cd
Let Mm,n = Mm,n(C) denote the m x n matrices over C. In the case that m = n,
we write Mn instead of Mn,,. Let Mn(Cd) denote dtuples with entries from Mn. Thus, an
X e Mn(Cd) has the form X = (X1,..., Xd) where each X, e Mn.
A noncommutative set is a sequence ((n)) where, for n e N, (n) c Mn(Cd),
which is closed with respect to direct sums; i.e., if X e (n) and Y c L(m), then
X Y = (X1 Y1,..., Xd Yd) E (n + m) (2.1)
where
xi (= x
A noncommutative set = ((n)) is open (closed) if each (n) is open (closed).
A matrix convex set C = (IC(n)) is a noncommutative set which is closed with
respect to conjugation by an isometry; i.e., if a c Mm,n and a*a = In, and if X =
(X ..., Xd) c C(m), then
a*Xa = (a**Xia, ..., a*Xda) e /C(n). (2.2)
A subset U of Mn(Cd) is circled if eieU c U for all 0 e R. A matrix convex set CK is
circled if each IC(n) is circled. As a canonical example of a circled matrix convex set,
suppose 7 > 0 and consider the noncommutative yneighborhoodC, = (C,(n)) of
0 e Cd defined by
d
C(n) = {X e Mn(Cd) XjX 72}.
j1
A matrix convex set /C = (C(n)), is said to be bounded if there exists 7 < 1, F > 1
such that, for each n c N,
C(n) C /C(n)C Cr(n), (2.3)
Assumption 2.1.1. Here, it is typically assumed that the matrix convex set /C
(a) is open;
(b) is bounded; and
(c) is circled;
2.1.1 Examples
The following are some examples of matrix convex sets in Cd that satisfy the
conditions of Assumption 1.
(i) Let /C(n) = {(X,,..., Xd) : IXJ < 1} with 7 = 1 and F = vd. /C = (/C(n)) is the
ddimensional noncommutative matrix polydisc.
(ii) Let /C(n) = {X = (X11, X12..., XdX) : IXop < 1}, where Xop is the norm of the
operator X = (X,)d 1 (Cn) (C)d, with 7 and F /C (n) is
the d x d noncommutative matrix mixed ball.
(iii) Let C(n) = {(X ,...,Xd) : sup{ d (X/y, y) : y 1} < 1} with 7= and
F = 2 d. If d = 1, then C is the collection of all strict numerical radius contractions.
2.1.2 Properties
Below we present three important properties of a matrix convex set = ((n)) in
Cd which we will use in the forthcoming chapters.
(i) (n) is convex for each n c N.
(ii) If 0 e L, then L is closed with respect to conjugation by a contraction; i.e. a in
equation (2.2) can be assumed to be a contraction.
(ii) The closure of L namely = ((n)) is matrix convex.
Proof. (i) Let X, Y e IC(n) and 0 < t < 1. Consider the the block matrix a* =
( tin 1 t) e Mn,2,. Since a is an isometry it follows that,
tX +(1 t)Y = *(X E Y)a IC(n).
(ii) Let a e Mn,, be such that Ij < 1 and X e (n). We first observe that
a = (a 0) e Mmn,.m,, is a contraction. Let
X = (X E 0) e C(n + m)
and
X = (X e 0) /C(2(n
m)).
Let Da denote the defect operator of 5. i.e.,
D = (Im +n )'.
Then the Julia matrix of a namely,
d Dj
Ji(s =Tr e
Da d*
is unitary. Therefore,
d*Xa = lm+n Om
It follows that
a*Xa =X Inm Omn)
(iii) Consider Mn(Cd) with the topology defined by the norm
11 l(X ... X ) d) l
(2.4)
mn /C(m
Om nJ
j=1
) J(5)*kXJ( 5)
X5 O e /C(m).
0n,m
where the norm on the RHS is the usual operator norm.
Let X = (X, ..., Xd) e (p) and Y = (Y1, ..., Yd) e (q). Choose sequences
Xm = (Xmi ... Xmd) and Y, = (Y,,1..., Ymd) from (p) and (q) respectively such that
Xm X and Y,+ Y.
We have,
d
II ( ,X Y) =  (X,, Y,,) (Xj Y Y,)
j= 1
d
S(Xr Xj) E (Yrj Yj)
j 1
d
= max{X,,j Xj ,  Yj  }
j 1
Since X, X and Yr, Y, the sum on the RHS above can be made arbitrarily
small for all large values m. Thus X Y c (p + q).
If a e Mp is an isometry, then we have
d
 aX,o a*X(,  = oa*(X,, X))o
d
j 1
Since X, X, it follows that a*Xoa (e).
2.2 Matricial HahnBanach Separation
In this section we present a HahnBanach separation theorem for matrix convex
sets in Cd due to Effros and Winkler (See [13]). The following contents, are minor
variants of lemmas and theorems from [16].
Given a positive integer n, let E, denote the collection of all positive semidefinite
n x n complex matrices of trace one. A T e E corresponds to a state on M,, via the
trace,
Mn A 3A tr(AT).
An affine linear mapping f : T IR is a function of the form f(x) = af Af+(x),
where Af is linear and af e R.
Lemma 2.2.1. Suppose F is a cone of affine linear mappings f : T, R. If for each
f e F there is a T E 7E such that f(T) > 0, then there is an S E 7E such that f(S) > 0
for every f c F.
Proof. For f e F, let
Bf = {S T: f(S) > 0}.
By hypothesis each Bf is nonempty and it suffices to prove that
nfEBf # 0.
Since each Bf is compact, it suffices to prove that the collection {Bf : f e F} has
the finite intersection property. Accordingly, let f, ...f, f F be given. Arguing by
contradiction, suppose
qn if 0.
In this case, the range F(T7) of the mapping F : T RW" defined by
F(S)= (fl(S),..., fm(S))
is both convex and compact because En is both convex and compact. Moreover, it does
not intersect
R" { = {x = (x, ..., x) : xj > 0 for each j}.
Hence there is a linear functional A : R"' R such that A(F(E)) < 0 and A(R"n) > 0.
There exists Aj such that
m
S(x) Ajxj
y 1
Since A(Rm) > 0 it follows that each Aj > 0 and since A / 0, for at least one k, Ak > 0.
Let
m
j= 1
Since F is a cone and Aj > 0, we have f e F. On the other hand, if T e En, then
f(T) < 0. Hence for this f there does not exist a T e n such that f(T) > 0, a
contradiction which completes the proof. O
Lemma 2.2.2. Let C = (C(n)) denote a matrix convex set in Cd which contains 0 e Cd.
Let n c N and a linear functional F : Mn(Cd) C be given. If
Re F(C(n)) < 1,
then there exists S E En such that for each m c N, Y e C(m) and C c Mm,n
Re F(C*YC) tr(CSC*).
Proof. For each m c N, Y e C(m) and C c M,,n, we define the affine linear map
fyc : n IR by
fyc(T) = tr(CTC*) Re F(C*YC).
Let Fn = {fy,c : Y C(m), C e M,,n, m e N}. The set Fn is a cone since
f3y,c = fy, c
and
fry,C, + fY2,2 = fZ,D
where Z = Yi Y2, D* is the block matrix (CQ C) and 8 > 0 is arbitrary.
Choosing T = aa* where a is a unit vector such that
II Cal =  C,
it follows that
fyc(7) = IC Re F(C*YC).
If II CII = 1, then by property (iii) from Subsection 2.1.2, C* YC e C(n) and so by the
hypothesis of the lemma, the right hand side of the above equation is nonnegative. If
C does not have norm 1, but is not zero, then a simple scaling argument shows that
f,c(T)> 0.
Hence by Lemma 2.2.1, there exists an S e Tn such that fy,c(S) > 0 for every m,
Y C(m) and C e Mm,n. O
A linear pencil L of size n is a formal expression of the form yL 1 Lege where
L, e Mn. For a dtuple T = (T,,..., Td) of bounded operators on a Hilbert Space TH, the
evaluation of L at T is defined as the operator L( T) = e Le Te.
Theorem 2.2.1 (Matricial HahnBanach Separation). LetC = (C(n)) denote a closed
matrix convex set in Cd which contains a noncommutative neighborhood of 0 e Cd. If
X C(n), then there is a linear pencil L (of size n) that satisfies the following conditions.
(i) 2 L(Y) L(Y)* > 0 for all m E N and Y e C(m)
(ii) 2 L(X) L(X)* E 0.
Proof. By applying the usual HahnBanach separation theorem on the closed convex
subset C(n) of Mn(Cd) and using the assumption that C(n) contains a noncommutative
neighborhood of 0, we obtain a linear functional F : Mn(Cd) + C such that
Re F(C(n)) < 1 < Re F(X).
Choose 0 < c < 1 sufficiently small such that G = (1 c)F satisfies
Re G(C(n)) < 1 < Re G(X). (2.5)
From Lemma 2.2.2 there exists S e Tn such that
Re F(C*YC) < tr(CSC*)
for each m c N, Y c C(m) and C c Mm,n.
If we let R = (1 e)S + then R Ec T, it is invertible and
If w letR = L )5 +n n
Re G(C*YC) < tr(CRC*).
(2.6)
Let {el,..., ed} denote the standard orthonormal basis for Cd. Given 1 < < d, and
column vectors c, d e Cn, define a bounded sesquilinear form on Cn by
B,(c, d) = G(R dcTR& e)
where cT denotes the transpose of c.
There exists a unique matrix Be e Mn such that
B(c, d) = (Bec, d).
Define the linear pencil L by dE BEg.
Fix a positive integer m. Let Y =(Yi,..., Yd) e C(m) be given and consider L(Y),
the evaluation of L at Y. Let {e, ..., em} denote the standard orthonormal basis of Cm.
For 6 = Ej1 J 0 ej e C" C"m, we have
m d
(L(Y)6,} = ~Bj, {Yej, e
ij=1 = 1
m d
1
= G(Ri'6JR ,
ij= 1 =i 1
m d
= G((R 2j) (RP
ij=1 = 1
d m
= G( (C(Ri)(Ye4,
e=1 ij=1
i G 1 
= G((R6)Y(R6)*)
ee)(Y ej, ei)
)0e* e)<(Y ej, ei)
ei)(R'6j)*) & e)
where J6 is the column vector whose entries are complex conjugates of the column
vector 6, and 6 is the n x m matrix with jth column 6j. Using equation (2.6) we get,
Re (L(Y)6, ) = Re G((R'6)Y(R6)*)
< tr((R'6)* R(R2)6))
m
11
= 'R(R6_), (R'6j))
j= 1
m m
=<( ;& e,, 0& ej)
i=1 j=1
= 116112.
On the other hand, computing as above and using equation (2.5) we get,
n n
Re (L(X) Rei, e,, Rej 0 ej) = Re G(InXIn)
i=1 j=1
>1
n
= IRei ei, 2.
i= 1
2.3 The Noncommutative Fock Space and Creation Operators
The Fock space and the Creation Operators that act on it play a central role in the
analysis to follow in the forthcoming chapters. One of the key properties is that tuples
of finitedimensional compressions of the creation operators lie in the underlying (open,
circled and bounded) matrix convex set IC. We provide a proof of this fact in this section.
2.3.1 The Free Semigroup on d Letters and Intial Segments
The Fock space is defined in terms of the free semigroup on d letters.
Let Fd denote the set of all words generated by d symbols {g, ..., gd}. Define the
product on Fd by concatenation. i.e., if w = gi...gi and v = gj1...gj,, then the product
wv is given by g,i...ggj,...gj,. Fd is a semigroup with respect to this product, with the
empty word 0 acting as the identity element, i.e. w0 = w = Ow for all w e Fd.
The length of the word w = gi,...gi, is declared to be m (where it is assumed that
g, / 0) and is denoted I w. The length of 0 is zero.
A set A c Fd is an initial segment if its complement is an ideal in the semigroup Fd;
i.e., if both gjw, wgj e Fd \ A (1 < j < d), whenever we Fd \ A. In the case that d = 1 an
initial segment is thus a set of the form {0, g,, g',..., gm} for some m.
2.3.2 The Noncommutative Fock Space
Let C(g) C(g, ..., gd) denote the algebra of noncommuting polynomials in the
variables {g, ..., gd}. Thus elements of C(g) are linear combinations of elements of Fd;
i.e., an element of C(g) of degree (at most) k has the form
k
j=0 Iwl=j
where the pw are complex numbers.
To construct the Fock space, F2, define an inner product on C(g) by defining
0 if w / v
(w, v} = : (2.7)
1 if w = v
for w, v e Fd and extending it by linearity to all of C(g). The completion of C(g) in this
inner product is then the Hilbert space F2.
2.3.3 The Creation Operators
There are natural isometric operators on F2 called the creation operators which
have been studied intensely in part because of their connection to the Cuntz algebra [7].
Given 1 < j < d, define Sj : F2 F2 by Sjv = gjv for a word v e d and extend Sj
by linearity to all of C(g). It is readily verified that Sj is an isometric mapping of C(g) into
itself and it thus follows that Sj extends to an isometry on all of F2. In particular S Sj = I,
the identity on F2. Also of note is the identity,
d
i~S SjS = P, (2.8)
j= 1
where P is the projection onto the orthogonal complement of the onedimensional
subspace of F2 spanned by 0, which follows by observing, for a word w e Fd and
1
S*(w) if w = gjv
0 otherwise.
Of course, as it stands the tuple S =(S1, ...,Sd) acts on the infinite dimensional
Hilbert space F2. There are however, finite dimensional subspaces which are essentially
determined by ideals in Fd and which are invariant for each S*.
The subset A(f) = {w : wl < } ofFd is a canonical example of a finite
initial segment. And the subspace F(T)2 of F2 spanned by A(f) is invariant for S5,
1 < j < d. Let V(f) denote the inclusion of F(f)2 into F2 and let 5() denote the operator
V(e)*SV(e). Thus, 5() = ((S())1,..., (S(f))d) where (S(C))j = V()*SjV(C).
Recall 7 from the definition of the (open, bounded and circled) matrixconvex set /C.
Lemma 2.3.1. If t < 7, then tS(f) e IC(n) for some n c N.
Proof. Let P denote both the projection of F2 and F(C)2 onto the orthogonal complement
of the span of 0 in F2 and F(C)2 respectively. It follows from equation (2.8) that
P =V()*PV()
=v(ef*Y Sisj* v(
d
Y= (Sv))(Sv));.
j=1
Thus for t < 7, that tS() e C,(n) c /C(n), where n = o= di is the dimension of
F()e2. D
Remark 2.3.1. S()W = 0 for all w A(C).
CHAPTER 3
ABSTRACT OPERATOR ALGEBRAS
We begin this chapter with the definition of an abstract operator algebra. Following
that we present some examples and the proof of the fact that the quotient of an
abstract operator algebra by a closed twosided ideal is an abstract operator algebra.
Furthermore, we present a characterization of an abstract unital operator algebra due to
Blecher, Ruan and Sinclair.
3.1 Abstract Operator Algebra
Let V be a complex vector space and Mp,q(V) denote the set of all p x q matrices
with entries from V. V is said to be a matrix normed space provided that there exist
norms I Ip,q on Mpq(V) that satisfy
IIA X BI,,r < IIAIIIIX IpqII BI
for all A Mc,p, X Mp,q(V), B Mq,r.
A matrix normed space V is said to be an abstract operator space if
IIX E Yp+e,4,, = max{Xll p,q,, I Yj,
where Xe Mp,,(V) and Y e M ,(V) and X Y =
v0 Y
It is important to note that, without loss of generality, we can replace the rectangular
matrices in the above definitions with square matrices.
V is an abstract operator algebra if V is an algebra, an abstract operator space
and if the product on V is completely contractive i.e. XYp < 1 whenever Xlp < 1
and I Yllp < 1 for all X, Y e Mp(V) and for all p. We say V is unitalif V contains a
multiplicative unit.
Before we look at examples of abstract operator algebras, we present, as a remark,
an interesting fact about the abstract operator space Cd.
Remark 3.1.1. The closed unit balls of the abstract operator space Cd form a matrix
convex set.
A partial converse to the above remark is the following.
Lemma 3.1.1. Let L = ((n)) be a closed, bounded, absorbing and circled matrix
convex set in Cd. If L is strongly circled, i.e. UL(n) c (n) for all n e N and unitary
matrices U e Mn, then there exists a sequence of norms II , such that L(n) is the
closed unit ball of Mn(Cd) with respect to II n, and Cd together with the sequence of
norms II n is an abstract operator space.
Proof. It follows from the hypothesis and the definition of matrix convexity that for any
unitary matrix U e Mn,
(n)U = U*(UL(n))U C (n).
This implies that for unitaries U, V e M,
ULn)V C (n). (3.1)
Since (n) is convex, closed, bounded, absorbing and circled, it is the closed unit
ball of M,(Cd) with respect to some norm, which we will denote I nII. We need to show
that Cd together with the sequence of norms II n is an abstract operator space.
Let X e M,(Cd) be such that IXn = 1 and A, B e M, be of unit norm. Consider
the Julia matrices (see (2.4)) J(A), J(B) of A and B. Since J(A) and J(B) are unitary, it
follows from equation (3.1) that,
IJ(A)(X O0)J(B)2n < 1.
Hence,
IIAXBEI =  In On J(A)(X 0)J(B) < 1.
If any of A, B c M, and X e M,(Cd) are not of unit norm, then a simple scaling
argument shows that IIAXBII < IAIIIIXlI B . Thus (Cd, 1 I n) is a matrixnormed
space. Next we show that if X e Mn(Cd) and Y e Mm(Cd) then, IIX YIIn =
max{X I Y m}.
Since (Cd, I In) is a matrix normed space it follows that,
IIX Yn m > max{lX,  Ym}.
To prove the reverse inequality, observe that II x IIn and YaxImII are
max{xll X ljlYll,.} max{f llXjll, Y }ma}
at most one. Hence, by the matrix convexity of L, it follows that,
X Y
max{llXlln, IIYI i, max{Xn, I IYIm})
3.1.1 Examples
(i) Let R be a Hilbert space and V = B(H). Define matrix norms on V by
i( (T ) p,q= IITII
where T is the operator (T,,)p, 1 : (q'T i c ) and I TI is its operator norm.
Then V together with the sequence of norms I Ip,q is an abstract unital operator
algebra.
(ii) Let R be an arbitrary separable Hilbert space, and V denote the algebra of
polynomials in d variables. Define matrix norms on V by
II(x,)lp,q = sup{(x, (T))ll}
where the supremum is taken over all dtuples T = (T, ... Td), where { Tk} = C
B(H) is a set of commuting contractions. Then V together with the sequence of
norms I IIp,q is an abstract unital operator algebra.
3.1.2 The Quotient Operator Algebra
Let V be an abstract operator space with the sequence of norms I Ip,q, and let W
be a closed subspace. Let r : V V/W denote the quotient map rT(x) = x + W. By
identifying Mp,q(V/W) with Mp,q(V)/Mp,q(W) we get a sequence of norms 1 IIp,q on
Mp,q(V/W) defined by
IIl(r(x,))ll l,q = inf{(xj + Yij), : yY E W }
Lemma 3.1.2. V/W with the sequence of norms    p,q defined as above is an abstract
operator space.
Proof. Let A c MKp, X = ( (xy)) e Mp,q(V/W) and B e Mqr. Choose (yi) E Mp,q(W)
such that
IIl( x ))lll p,q + > II(x, + y )lp,q. (3.2)
Observe that
AXB = A((x)) B = I(A(x))B) = T(A(x + y)B). (3.3)
Using equations (3.2) and (3.3) gives,
IIIAXBIII ,q = I I(A(x + y,)B))  Ip,q
< IIA(x + yj)BIp,q
< IIAII (x,+ Y ) p,q)II BI
= IIA II(IIeIX II l,, c)IIB II.
By letting c 0, it follows that V/W is a matrixnormed space.
Let X = (q(x,)) e Mp,q(V/W) and Y = (q(ymn)) E Mt.,(V/W). Since V/W is a
matrixnormed space, it can be seen that
IIIX E Ylp+,q > > max{Xlp,q, II Yl,,}.
To prove the reverse inequality, choose (a,) e Mp,q(W) and (bmn) e Mr.,(W) such that
11l(Tx( ))lll,. + > II(xd 4+ ad)lp,q (3.4)
IIl(T (ym))ll + > (ymn bmn),, (3.5)
and observe that
X Y Y = n((x, ay a) (ynn + bmn)). (3.6)
Using equations (3.4), (3.5) and (3.6) yields,
X E Yl ,p+t, = II Ir ((xj + ay) ( (y n + bmn))l Ip+,q+r
< I(xy + ay) (ymn + bmn) p+,q+ r
= max{(x + ay) p,q, I(Ymn + bmn).e),r}
< max{X p, III Ylll ,} + .
Letting c 0 completes the proof. O
Corollary 3.1.1. Let V be an abstract unital operator algebra and W be a closed
twosided ideal in V. Then, V/ W is an abstract unital operator algebra.
Proof. From Lemma 3.1.2 we know that V/W is an abstract operator space. Moreover,
1 + W is the unit of V/W. It remains to show that the product on V/W is completely
contractive. For that purpose, let X = (r(x,()) and Y = (r(yy)) e Mp(V/W). Choose
(ay), (bu) e Mp(W) such that
IiI l(s xj))lll p + > I(x o+V adt)l op (3.7)
l(T(y ))lllp + e > (yu + b )llp (3.8)
Since W is a twosided ideal of V, it follows that
XY = (Tl(xy))(Tl(yy)) = Tl((X)(y)) = l((xu + a)(yyi + by)) (3.9)
Using equations (3.7), (3.8), (3.9) and the fact that multiplication in V is completely
contractive yields,
IIIXYlllp = II((xy ay)(yy + b,)) lp
II(xdj + ad)(yd + b,)llp
"< I(x + a,jlp\l(y + b,)llp
< (lll(x ))lllp + )(Y IIIP + lll( Y))lll )
The corollary follows by letting c 0. O
3.2 Representations of Abstract Unital Operator Algebras
The following is a characterization of abstract unital operator algebras due to
Blecher, Ruan and Sinclair.
Let V and W be abstract operator spaces and : V V W be a linear map. Define
Oq: Mq 0 V  Mq 0 W by Oq = Iq & where Iq is the q x q identity matrix.
The map 0 is said to be completely contractive (isometric) if Oq is a contraction
isometryy) for each q e N.
A completely contractive (isometric) representation of an algebra A is a completely
contractive (isometric) algebra homomorphism 0 : A , B(M) for some Hilbert space
M.
Theorem 3.2.1. (BlecherRuanSinclair) Every abstract unital operator algebra A
admits a completely isometric representation. i.e. there exists a Hilbert space M and a
unital completely isometric algebra homomorphism 0 : A B(M)).
CHAPTER 4
THE ABSTRACT OPERATOR ALGEBRAS A(/C)" & A(1C)"/I(kC)
Recall IC, the matrix convex set satisfying the conditions of Assumption 2.1.1. In
particular, IC(1) is an circled open convex subset of Cd. This chapter is divided into two
parts. In the first part, which consists of four subsections, we construct an abstract unital
operator algebra which is a natural noncommutative analog of the Banach algebra
H(IC(1)). We also present a few lemmas on completely contractive representations of
this algebra.
In the second part, Sections 4.5 and 4.6, we consider the ideal Z(/C) of the algebra
A(KC)" determined by a finite initial segment A. We show that the quotient algebra
A(IC)"/Z(/C) determined by the ideal is an abstract unital operator algebra. We also
show that norms of classes in the quotient algebra are attained.
4.1 The Algebra A(KC)" of Scalar Formal Power Series
In this section we establish that the collection of scalar formal power series in
noncommuting variables which converge uniformly on K is an algebra. We begin with
the definition of a formal power series.
4.1.1 Formal Power Series
Let U and U' denote separable Hilbert spaces. A formal power series with
coefficients from B(U, U') is an expression of the form
S fw (4.1)
wEEd
where fw e B(U,U'). It is convenient to sum f according to its homogeneous of degree
terms; i.e.,
f = fww = (4.2)
j=0 Iw=j j=0
Recall, for a dtuple T = (TT,..., Td) of operators on a common separable Hilbert
space TH and a word w = ggg,... d, i {1, 2 ..., d}, the evaluation of w at T
is defined as
T = T T, ... T .
Given a formal power series f as above, define
00
f(T)= fw DTW (4.3)
j=0 Iwl=j
provided the sum converges in the operator norm in B(U X, U' 'H) in the indicated
order. We note for clarity that if U = U' = C, then 0 in (4.3) is the usual scalar product
and if f(T) converges, it is an element of B().
Recall the matrix convex set K = (IC(n)) which satisfies the conditions of
Assumption 2.1.1. We will write X e KC to denote X e UnN I(n). For the formal
power series f as above, we define
If l = sup{ f(X)  : X /C}. (4.4)
4.1.2 The Vector Space A(/C)"
As it stands, the supremum in equation (4.4) can be infinite. We are only interested
in those formal power series f for which this is not the case. Let
A(IC) = f f= W:f fw e C, f < oo.
S wEFd
Thus, elements of A(/C)" are in some sense analogous to elements of the classical
commutativee) Hardy space, H"(/C(1)) of bounded analytic functions on /(1). It is not
hard to see that A(/C)") is a complex vector space with respect to termwise addition
and scalar multiplication.
Lemma 4.1.1. 1  defines a norm on A(1C)".
Proof. It follows from the definition that I  is a seminorm. Thus it suffices to show that
 fl = 0 implies f = 0. Let e { 0, 1, 2,... } and 5(f) be as in Subsection 2.3.3. For
0 < t < 7, using Lemma 2.3.1 and Remark 2.3.1, we get
0 = Ilf2 (4.5)
> f(tS( ))(0) 2 (4.6)
= tj fW2 (4.7)
j=0 Iwl=j
= t2 f .2 (4.8)
j=0 w =j
Thus fw =0 for all w such that Iwl < Since f is arbitrary, the lemma follows. O
4.1.3 Matrix Norms on A(KC)"
Since it will be necessary to consider, in the sequel, matrices with entries from
A(KC)", we define them here. Let
Mp,q(A(C)")= f fww: fw Mpq, ,,fp.,<
wEFd
where the norm I Ip,q is given by
Ifllp, = sup{f(X) : X e KC}. (4.9)
The following Lemmas plays an important role in the analysis to follow generally,
and in proving that A(KC)" is an algebra, in particular.
00
Lemma 4.1.2. Iff = fww e Mp,q(A(/C)0), then 2 If2 converges.
wEFd j=0 Iw=j
Proof. Recall 7 from equation (2.3) and S() from Subsection 2.3.3. Fix 0 < t < 7 and a
unit vector x e Cq. Using Lemma 2.3.1 and Remark 2.3.1, we get
j=0 wl=j
J=0
>  t
j=o
= t
j0 I
0)112
S()w (x 0)112
) w112
Since E is arbitrary, allowing t T 7 yields
(4.10)
j= fwx2 <
j=o Iwl=
Let {el, e2...., eq} denote the standard orthonormal basis for Cq. We know that for
each we cFd
fI < I fee .2
i 1=
(4.11)
Moreover, for each 1 < i < q, equation (4.10) implies that
Using equations (4.11) and (4.12), we get
o
2= j 2 i IfWI2
j=0o Iwl=j
oo q
< 72i Y'(Y Ifweell2)
= (  fe, l2)
< qlwfl i2.
< q f 2
5fw2
WI j
WI j
wl=j
2  fwee 2 < f 2.
j=0 wl=j
(4.12)
(ts(V))w(x
Lemma 4.1.3. Suppose that f = fww e Mp,q(A(KC)") and X e IC(n). Let
weE
A,= f XW.
If < r < sup{s > 0 : sX e /C(n)}, then
r llA l < I fll .
In particular, there is a p < 1 such that IIAj II < pl I f l.
Proof. Because IC(n) is open, convex, and circled, the function F(z) = f(zX) is defined
on a neighborhood of D; i.e. there exists a 6 > 1 for which the series,
00
F(z) = Ajz
j=0
converges for all z such that z < 6. Using the fact that the series F(z) converges
uniformly on the closed disc {z : zl < (} for every ( < 6, we get, choosing 1 = 1, for
each j that,
1 02T
271 0
It follows that
A < 2F(eit) dt.
Since I F(e't) =  f(e'tX) and eitX e IC(n), it follows that IF(e't)l < f f and the
lemma follows. O
4.1.4 The Algebra A(/C)"
There is a natural multiplication on A(/C)" which turns it into an algebra over C.
Given f = wE,, fww e Mp,(A(/C)00) and g = wE gww e Mq,,(A(kC)"), define
the product fg of f and g as the convolution product; i.e.,
fg9 = ( fu"v ) W.
wE)7d \uvI w /
The remainder of this subsection is devoted to demonstrating that this convolution
product corresponds to pointwise product, extends the natural product of noncommutative
polynomials (formal power series with only finitely many nonzero coefficients), and
makes A(KC)" an algebra with unit 0.
Lemma 4.1.4. Iff C Mp,4(A(/C)") and g e M,,(A(/C)") and X e /C, then
(i) fg(X) converges;
(ii) fg(X) = f(X)g(X);
(iii) fg is in Mp,r(A(KC)"); and
(iv) Ifg < f Ifl lgll.
Proof. Fix X e IC(a) c KC.
(i) As in the proof of Lemma 4.1.3, let
A,= f, X",
Iwl=j
Observe that C = ECo AkBjk
Let F(z) = f(zX) and G(z) = g(zX), both of which are defined in a neighborhood
of D. From Lemma 4.1.3, there is a p < 1 such that iAmr\ < prllfll and IIBk < pkllgll.
Hence
j=0 k=0
converges. In particular fg(X) = C0o Q converges in norm.
(ii) Consider the function FG(z) = fg(zX). From the proof of (i) we know that FG(z)
is defined whenever z < We will prove the more general fact that FG(z) = F(z)G(z)
whenever z < from which the claim will follow by setting z = 1. Recall 7, F from the
definition of the matrix convex set IC. Let z be such that z < 2. Observe that for any k,
I C zj A,,zm Bz"  < C z Azm B zn
j= 0 m= 0 /n=O j=0 m= 0 n=0O
k oo
 Amzm 1111 Bnz 
m=0 n=k+l
oo k
II 1 Amzm llll 11 B
m=k+l n=0
 amzmllll Bz"11
1 Amz BnA z n
m=k+l n=k+l
(4.13)
We claim that the LHS of equation (4.13) converges to zero as k oo. It suffices
to show that the first term on the RHS of equation (4.13) converges to zero, in view of
the convergence of the second, third and the fourth terms on the RHS to zero due to the
following reasons:
(a) both k AmZ'm and I EC 0 Bnz are finite.
(b) both 1Ek+1 ArmZm and k+ 1 BZn, being tails of the convergent series f(zX)
and g(zX) respectively, converge to zero as k oo.
Consider the first term on the RHS of equation (4.13).
k k ( z\
mz"' Bnzn
m0 (n0
(fgv XU V)
k
(fugv XUV)
k
, (f gv)
k,lvl=k
+
ul= 2,vl k1
+
ul=3,vl=k1
x"u) IZ2k
(fgv XUV) ...
(f.gv0 XV)+...
S(fgv XuV) I Zlk 1
uI k,lvl 1
S(f9gv XUV) Izlk+2
ul k,lvl 2
< ( Ifugv11
Jul 1,vl =k
lul=2,vl=k1
( 5 fgvll + I f 1gv
ul=2,vl=k ul=3,vl= k1
.. ( I fgv (Fz)2k
ulk,lvl =k
...+ f vf l (Flz)k 1
u =k, lv=1
S...+ I fgv (Fz)k+ 2
uk,lvl2
<( u I 2 kg 1 ,2 )(d k 2
Iwlk +luv w w 2uv=w
+ ....  If 11 v 1l
w=2k uv=w
< ( 4(k+1) iif. (kd +2)dk+ 1
11 2 2dk+l +
ww =k+1 uv=w
+ .4(2k) iif. fv 2(2k )d2k
2"d2k
Iw\=1k uv w
S ,4(k 2) I fgv) 2 (kdk +2
w k 2 uvw+2
wlk+2 uI w
=o
j=
+ 111
u =1 v=
ul =2, 1v
+.. + 11
IL11
IIfu9gv
k uv=w
(2 d2k
7+ 2m + 4 k+2) f
J dk 1 d d1k
Iwl= uv=w
Since the series d ... d are convergent,the
k=0 k=0 k0
corresponding kth term sequences converge to zero, and so does the sum of the
sequences namely ... / Moreover Lemma 4.1.2 implies
that e g2m ro fie2 and t t2n" 5 gvl 2 are finite. These facts together imply the
desired convergence. Thus fg(zX) = f(zX)g(zX) whenever z <
Fix x e eris and y e CP Ca and consider the complex valued functions
A(z) = (cG(z)x, y) and B(z) = (F(z)G(z)x, y). Observe that A(z) and B(z) are
analytic on the disc {z : Izl < } and A(z)= B(z) on the subdisc {z : zl < }. Hence,
A(z) = B(z) on {z : Izl < ,}. Since x and y are arbitrary, we have FG(z) = F(z) G(z)
whenever znl < Choosing z = 1 gives fg(X) = f( X)g(X).
(iii) & (iv) Since, for each X E C, fg(X) = f(X)g(X) it follows that fg(X) <
fl Ig. Thus Ifgl < fl Igl and fg e Mp,r(A(C)"). D
Corollary 4.1.1. A(/C)" is an algebra.
Proof. Take p = q = r = 1 in the above lemma. O
4.2 Weak Compactness and A(KC)"
In this section it is shown that every bounded sequence in A(KC)" has a pointwise
convergent subsequence. Indeed, A(KC)" has weak compactness properties with
respect to bounded pointwise convergence mirroring those for H"(D), the usual space
of bounded analytic functions on the unit disk D.
Proposition 4.1. Suppose that fm = wCe (fm)ww is a Mp,,(A(/C)) sequence. If,
for each X e 1C the sequence (fm(X)) converges or if for each w E Fd the sequence
(fm)w converges and if (fm) is a bounded sequence (so there is a constant c such that
II fml < c for all m), then there is an f e Mp,,(A(/C)") such that fm(X) converges to f(X)
for each X e 1C and moreover I fll < c.
Proof. Recall S(f) defined in Subsection 2.3.3. Fix 0 < t < 7. If fm converges pointwise,
then the sequence (fm(tS(f))) is Cauchy. Given e > 0, there exists N E N such that
Ilfm(tS()) f(tS(t))l <
for all m, n > N. Thus, if x e Cq be a unit vector, then
e2 > Ifm(tS()) f,(ts())l2
> I f(tS(f)) f(tS(f))(x 0) 2
= t2J ((fm)w (fn)w) (x) 2
= o Iwl=J
> t2ll((fm)w (f)w) (x) 11
for each word w of length at most Since f and the unit vector x e C are arbitrary, it
follows that, the sequence ((fm)w) c B(Cq, CP) is Cauchy for each word we Fd. Thus
(fm)w converges to some fw for each w.
Hence, to prove the Proposition it suffices to prove that, if (f,), converges to f, for
each w and IIfmll < c for each m, then for each X e /C, the series
00
f(X)= f, O XW
j= o w =
converges and (f,(X)) converges to f(X).
For each j and X e C,
S (fm)w XW fw XW. (4.14)
Iwl=j Iwl=j
From Lemma 4.1.3, there is a p < 1 such that for each j,
II (fm)w, XW
Iwl=J
From equations (4.14) and (4.15), it follows for each j that,
II fw, XWl < c,
Iwl=J
an estimate which implies that the series f(X) converges.
Fix J e N such that
< (4.16)
4c
j=J+1
Recall F from the definition of the matrix convex set /C. Choose K e N such that for
all m > K,
ll( ) fwll < 2(J(4.17)
2(+ J 1)J +
for each 0
Thus for all m > K, from equations (4.16) and (4.17), it follows that
Ilfm(X) f(X)ll
<  1 (( M)w
j=0o Iwl=
j=0 Iwl=j
o0
fw) XW" 1
j=J+1
fl r ~j I (m)w
j J+1 Iwl
_/^j+i M/^_
J 00
2(J 1) 2c p
j=0 JJ1
S.
Thus fm(X) f(X) for all X c /C. Since fm(X)ll < c, we have If(X)ll < c. This
implies that Ilfll < c.
Lemma 4.2.1. If fm
Ewe (fm,)wW C Mp,q(A(/C)") satisfies Ilfm < c for allm c N
then,
(i) I(fm)wl < c forallw e Fd and forallme N;
(ii) There exists a subsequence {f,,} of{f,} and fw e Mp,q such that (fm,)w f for
all w;
(iii) Let f = f w. For each X c IC the sequence (fi, (X)) converges to f(X) and
wMeE
moreover l f(X)II < c.
Proof. To prove item (i), Recall 7 from the definition of /C. Fix 0 < t < 7 and a unit vector
x e Cq. Forj
0, 1, 2, ..., the hypothesis I fmll < c together with the conclusion of
Lemma 4.1.3 for X = tS(f) imply that
 t (f=)j
w=j
II ((fm)
wl=j
f,) XWII
xWII
xWII
JJ1
II fw
I j
s()w < c.
Hence
c2 > t (fm,),x S()wO 12
Iwlj
St2J I (fm)wX112
Iwlj
> t2, Jll(fm)wX 112
Since x and f are arbitrary, letting t 1 7 it follows that (fm)wl < < for all m e N.
The proof of item (ii) uses a standard diagonal argument. Let {wl, w2, ...} be an
enumeration of words in d which respects length (i.e., if v < w, then vl < Iwl). Since
I(f,)w, 1 < c there exists a subsequence say, {f,,m} of {f,,} such that (fi,m)w, fw,.
Since 1(fim)w2 < c, there exists a subsequence say, {f2,m} of {fi,m} and thereby of
{f,}, such that (f2,m)w2 f2. Continue this procedure to obtain a subsequence {fk,m} of
{fk1,m} and thereby of {fm} such that for all k e N,
(fk,m)w fwk
Now consider the diagonal sequence {fm,m}. It follows that {fm,m} is a subsequence
of {f,} and satisfies (f,,m)w fw for all we c d.
In view of what has already been proved, an application of Proposition 4.1 proves
item (iii). D
4.3 The Abstract Operator Algebra A(KC)"
Consider A(KC)" with matrix norms I p,q on M, q(A(/C)") as defined in (4.9).
Theorem 4.3.1. A(KC)" with the family of norms II IIp,q, is an abstract unital operator
algebra.
Proof. Let A c M.,, F c Mp,(A"), B e Mq,,. Interpret A and B as AO e M.,p(A(/C)")
and BO e Mq,r(A(KC)") respectively. As a notational convenience we will drop the
subscripts that go with the norms. It follows from Lemma 4.1.4 (ii) that for all X e IC(n),
IIAFB(X)II = IIA(X)F(X)B(X)II < IIA & I, F(X) IB 0 Ini < IA IFI BI.
Thus,
IIAFBI < IA IFI BI. (4.18)
Let F e Me ,(A(/C)), G e Mp,q(A(/C)"), X e IC(n). Observe that
(F(X) 0N^
F G (X) = F(X) 0(X) < max{F(X),  G(X)} < max{ FII, GII}.
0 G(X)
Thus,
IF GI < max{F, IG} (4.19)
Let c > 0 be given. Without loss of generality assume that IF > I GI. Choose m e N
and R e IC(m) such that IF(R)II > IFII c. Therefore
F GI > ( ) ) > IF(R)I > F e. (4.20)
S0 G(R)
Letting c 0 in the inequality (4.20) and from the inequality (4.19) it follows that,
IF G  = max{F, IG}. (4.21)
Lastly, complete contractivity of multiplication in M,(A(KC)") follows directly from Lemma
4.1.4 (iv). Thus A(KC)" is an abstract operator algebra. O
4.4 Completely Contractive Representations of A(KC)"
Recall the definitions of a completely contractive and completely isometric
representation from Section 3.2. Theorem 3.2.1 guarantees the existence of a
completely isometric representation for the abstract unital operator algebra A(C)".
Let : A(/C)"  B(M) be a completely contractive unital representation and let
T = (T, ..., Td) where T, = (gj). As a notional device, we will write 7T for T. Further,
we will also use 7, to denote the map Iq 7 : Mq(A(lC)") Mq 0 B(M).
In this section, we prove that for a completely contractive representation 7r of
A(/C)"), for any n e N and finite dimensional subspace W of M of dimension n and
0 < t < 1 the tuple
tZ = tV*TV = (tV* T V,..., tV*TdV)
is in IC(n). The proof begins with a couple of lemmas. Given f e Mq(A()C)") and
0 < r < 1, define fr as follows. If
f= EEfw = (4.22)
j 0 Iwl= j=0
then
00 00
fr= ri fww = rj.
j=0 Iw=j j=0
Lemma 4.4.1. If 7T is a completely contractive representation of A(1C)0 and f e
Mq(A(1C)"), then fr(T) converges in operator norm. Moreover r(fr) = fr(T) and
IIf,(T)II < fIIrll < If ll. If in addition 7T is completely isometric, then limr fr(T)  =
Proof. Write f as in equation (4.22). Lemma 4.1.3 implies that 1~ fj1 < I f l. Because T7
is completely contractive I(T) < If l. It follows that fr(T) converges in norm. Since
also the partial sums of fr converge (to fr) in the norm of Mq(A()C)"), it follows that
ir(fr) = fr(T) and so If,(T) < Ifr.
The inequality Ifrl < lf ll is straightforward because rC c /C.
Now suppose that 7T is completely isometric. In this case  fr(T)  = I fr,. On the
other hand lim I fr, = Ilf D
Lemma 4.4.2. Given k x k matrices A1,..., Ad, let
d
j=1
Suppose
2 L(X) L(X)* > 0
for all X e IC(T) and for all e N. Let OL denote the formal power series,
OL = L(2 L)1 LJ1
2J1
j=0
(a) 2 L(X) L(X)* > 0 for all X e C if and only if 2 L(U) L(U)* > 0 forall
U eC.
(b) If X e IC(f), then OL(X) converges in norm; i.e., the series
L(X)+ 1
2j+1
j 0
converges.
(c) II1L(X)II < 1 and hence OL is in Mk (A(C)00) and has norm at most one.
(d) If ,T is a completely contractive representation of A(IC)0, then 2 (L(T) +
L(T)*) 0.
Proof. To prove part (a), suppose that 2 L(X) L(X)* > 0 for all X e /C. Since K
is the closure of KC it follows that 2 L(U) L(U)* > 0 for all U e kC. To prove the
converse, assume the contrary, i.e. suppose there exists X e K and a unit vector v such
that ((2 L(X) L(X)*)v, v) = 0. The following argument has been adapted from [16].
Define the map q : R  R by
q(t) = ((2 L(tX) L(tX)*)v, v).
Observe that the q is an affine map that satisfies q(0) = 2 and q(1) = 0. Hence q(t) < 0
for all t > 1. Choose s > 1 such that sX e /C. Such an s exists because K/ is open. For
this s, we get
q(s) = ((2 L(sX) L(sX)*)v, v) < 0
which is a contradiction to the hypothesis.
To prove part (b) of the lemma, let X e /C(f) be given. Because IC(f) is circled, it
follows that e'eX e /C() for each 0. Hence,
2 e'oL(X) eL(X)* > 0 (4.23)
for each 0. For notational ease, let Y = L(X). Thus Y is a k x kt matrix and equation
(4.23) implies that the spectrum of Y lies strictly within the disc; i.e., each eigenvalue of
Y has absolute value less than one. Thus,
2 = (2 Y)
j= 0
converges in norm. It follows that
) 2Jy+1
L(X) = Y(2 Y)= 2j+1
J=0
converges.
To prove (c) observe that I Y(2 Y)l11 < 1 if and only if
(2 Y)*(2 Y) > Y*Y
which is equivalent to 2 (Y + Y*) > 0. Thus IIL(X) < 1 which implies that
OL E Mk(A(/C)0) with 1L < 1. This completes the proof of (c).
To prove part (d), observe, Since ,7 is completely contractive and OL E Mk(A(/C))
with norm at most one, an application of Lemma 4.4.1 yields, IL(rT)II < 1. Arguing as
in the proof of part (b), it follows that 2 (L(rT) + L(rT)*) > 0. This inequality holds for
all 0 < r < 1 and thus the conclusion of part (c) follows. D
Proposition 4.2. If T = (T,,..., Td), and j B( 4M) for some Hilbert space M, and
r(gj) = T determines a completely contractive representation of A(1C), then, for each
positive integer n and finite dimensional subspace W of M of dimension n and each
0 < t < 1 the tuple
tZ = tV*TV = (tV* T V,..., tV*TdV)
is in 1C(n), where V : W M is the inclusion map.
Proof. Let n and H be given and define Z as in the statement of the proposition.
Suppose that L is as in the statement of Lemma 4.4.2. From part (d) of the previous
lemma, it follows that 2 (L(T)+ L(T)*) > 0. Applying Ik 0 V* on the left and Ik 0 V on
the right of this inequality gives,
2 (L(Z)+ L(Z)*) = (k 0 V*)(2 (L(T) + L(T)*)(Ik 0 V) > 0.
Part (a) of Lemma 4.4.2 and an application of Theorem 2.2.1 imply that Z e /C(n).
Hence tZ e 1C(n) for all 0 < t < 1. D
Lemma 4.4.3. Let A c d be a finite initial segment, f C Mq(A(/C)") be as in equation
(4.22) and suppose that T is a completely contractive representation of A(1C)" into
B(M) and T is A nilpotent. Then I fr(T)l < sup{ f(X) : X e /C, X is A nilpotent}
for all 0 < r < 1. Moreover if f = 0 for all w A, then If(T)  < sup{ f(X)  : X e
C, X is A nilpotent}.
Proof. Since A is finite and T is A nilpotent, fr(T) = wA f, 0 (rT)w. Let {ej}j 1
denote the standard basis of Cq. Given c > 0, choose a unit vector y = e; 0 hj e
Cq M A such that
IIf,(T) < If,(T)yll + .
Let H denote the finitedimensional subspace of M spanned by the vectors
{T"(h) w e A, 1 < j < q} and V : H M4 be the inclusion map. Then Z = V* TV is
Anilpotent and
V* TWV if w A
Zw ={
0 otherwise.
Proposition 4.2 implies that rZ e 1C. Thus,
lfr(T)ll < ( fw rlwI T" yll+
= fII(Z)yll+
< fr(Z) +C
< sup{ f(X)  : X e /C, X is A nilpotent} + .
Letting c 0 yields the desired inequality. If fw = 0 for all w A, then f = weA fwW is
a noncommutative polynomial in which case we have limr, Ifr(T)I = 11 f(T)II and this
completes the proof. D
4.5 The Abstract Operator Algebra A(kC)/IZ(C)
Recall the definition of an initial segment from Subsection 2.3.1. Fix a finite initial
segment A c Fd and let
Z(/C) = f= ffww : f < 0 c A(/C)"
Observe that Z(IC) is a twosided ideal in the operator algebra A(C)".
Lemma 4.5.1. Mp,q(Z(lC)) is closed in Mp,q(A(IC)").
Proof. Let fm = CwEA(fm)wW be a sequence in Mp,q(I(IC)) be such that fm f =
EW7dw fww e M,,q(A(C)"). We need to show that f = 0 for all w e A. Let f be the least
integer such that A c A(.). Consider tS(f) where 0 < t < 7. Given C > 0, there exists
N e N such that
fm(tS()) f(ts()) <
for all m > N. Let x E Cq be an arbitrary unit vector. We have
> I fm((tS()) f(tS(f))l
> (fm( tS()) f(tS ()))(x 0)) 2
IS t W((fm)w fw)X W W + WfWX DW 12
wEA()\A wEA
=  t2wI l((f,), f )xl 2 w t2wl 1 fwx 2
wEA()\A wEA
> t21WI fwx 12
for all w A and m> N. Hence fw =0 for all w A.
By identifying Mp,q(A((C)"/I(lC)) with Mp,q(A(1C)")/Mp,q(I(1C)), Corollary 3.1.1
implies that the quotient A(KC)"I/(/C) is an abstract operator algebra. We formally
record this fact.
Theorem 4.5.1. A(IC)"/I(/C) is an abstract unital operator algebra.
4.6 Attainment of Norms of Classes in Mq(A(C)C")/Mq(Z(/C))
Let p e Mq(A(/C)"). In this subsection it is shown that there exists f e Mq(I(/C))
such that
p + f = lp + M,(Z(IC))I = inf{p + gll g e M,(I(/C))}.
Let {fm} be a sequence in Mq(I(/C)) such that
p + M,(Z(IC)) < lp + fml < p+ M,(Z(IC))I +
It follows that the sequence {fm} is bounded and that p + fmll lip + Mq(Z(C)). An
application of Lemma 4.2.1 yields a subsequence {fmk} of {fm} and f e Mq(I(IC)) such
that
(p+ f)(X) (p + f)(X)
for all X E /C.
Proposition 4.3. If p, {fmk}, f are as above, then lip + f = lip + Mq(I(IC)).
Proof. Let e > 0 be given. Choose R /C such that
lp f < I(p+ f)(R) + (4.24)
4
Since I(p + f)(R) I(p+ f)(R) , there exists K1 e N such that,
I(p+ f)(R)ll < I(p + fm)(R) + (4.25)
for all k > K1. Combining the inequalities from equations (4.24) and (4.25), implies that,
for all k > K1,
lip + fl < lip fm, + (4.26)
Since lip + fm, II p + Mq(I(/C)) , there exists a Natural number K2 such that for
all k > K2,
lip + f, < lip + Mq(I(C))I + (4.27)
Setting k = max{Ki, K2} in equations (4.26) and (4.27), and letting C 0 yields
lip+ fll < lp+ M,(I(/C))I.
On the other hand, since f e Mq(Z(IC)),
lip+ fll > lp+ M,(Z(/C))I.
E
CHAPTER 5
THE NONCOMMUTATIVE CARATHEODORYFEJER PROBLEM
In this chapter we pose the CaratheodoryFej6r Interpolation problem (CFP) for our
open, bounded and circled matrix convex set /C. Using the results from Chapters 2 4,
we prove a necessary and sufficient condition for the solvability of the problem.
5.1 The CaratheodoryFejer Interpolation Problem (CFP)
The statement of the CFP is as follows: Fix a matrix convex set K/ satisfying the
conditions of Assumption 2.1.1. Let A c 'd be a finite initial segment, and
P = p,w A(/C)0
wEA
be given. Does there exist x E A(/C) such that xw = pw for w e A and I1 < 1?
Theorem 5.1.1. There exists a (minimumnorm) solution x to the above problem if and
only if
sup{p(X)l : Xe c X is A nilpotent} < 1.
The generalization of Theorem 5.1.1 allowing for operator coefficients is proved in
this chapter.
The strategy is to first prove the result for matrix coefficients. This is done in Section
5.2 below. Passing from matrix to operator coefficients is then accomplished using
wellknown facts about the Weak Operator Topology (WOT) and the Strong Operator
Topology (SOT) on the space of bounded operators on a separable Hilbert space. The
details are in Section 5.3.
5.2 The Matrix Version
Fix A c Fd, a finite initial segment, and a polynomial p = EwC, pww e Mq(A(/C)").
Proposition 5.1. There exists f e Mq(A(KC)") such that lip + fll = lp + Mq(I(/C)) =
sup{p(X)  : X IC, X is A nilpotent}.
Proof. From Theorems 4.5.1 and 3.2.1 it follows that there exists a Hilbert space
M and a completely isometric homomorphism 0 : A(C)"/IZ(KC) B(M4). As
before, identify Mq(A(/C)/Z(K/C)) with Mq(A(lC)")/Mq(IZ(l)). Let 0q denote the map
,Iq 0 : Mq(A(IC)f)/Mq(I(1C)) M, B(M.A). Let R be the dtuple (R1, R,,..., Rd),
where R, = 0(gj + Z(IC)) e B(M), for 1 < j < d. Observe that R is Anilpotent. Let
l : A(KC)" A(KC)"/IZ(K) be the quotient map. The composition 7 = 0 o r : A(C) )
B(M) is a completely contractive representation of A(KC)". Since r(gj) = Rj, consistent
with the notation introduced in Section 4.4, we will use rR to denote the map r.
It follows from Theorem 4.3 that there exists f e Mq(I(IC)) such that
lp+ fl = l p+ M,(I(/C)). (5.1)
The fact that 0 is completely isometric implies that
p + M,((I(C)) I = Il(P + M,(I(/C)) = Ip(R)II. (5.2)
Since 7R is a completely contractive representation of A(/C)", Lemma 4.4.3 implies that
Ilp(R)II < sup{p(X) : X e /C, X is A nilpotent}. (5.3)
Combining the equations (5.1), (5.2) and (5.3), it follows that
lp + fl < sup{p(X)l X e /, X is A nilpotent}.
But the definition of lp + f implies that
lip f+ > sup{p(X) : X e IC, X is A nilpotent}
and this completes the proof. D
(The matrix version of) Theorem 5.1.1 follows from the above proposition by setting
S= p f.
5.3 The Operator Version
As before, let A c Fd be a finite initial segment. Departing from the previous
section, let U be an infinite dimensional separable Hilbert space and let the polynomial
P = YW,, pw, where now {pw}wWE c B(U), be given.
Theorem 5.3.1. There exists a formal power series x = Cv,,e d w such that w = pw
for all w e A and Ix\11 = sup{lp(X)  : X e IC, X is A nilpotent}.
Proof. Let {ul, u2,...} denote an orthonormal basis for the separable Hilbert space U
and let Urn be the subspace of U spanned by the vectors {uj} 1. For notation ease, let
C = sup{lp(X)l : X e X is A nilpotent}. Observe that C = 0 if and only if p = 0.
For w c A, define Mm 3 (pm)w = V,~PwV where Vm : Um  U is the inclusion map.
Let pm denote the formal power series
Pm = Y(Pm)wW
wEA
For each X e IC, Observe that Ilpm(X)II < Ilp(X)II < lpll. Thus Ilpll < lpll and
pm E Mm(A(1C)) for all m c N. From Proposition 5.1, there exists fr, e M(Z(I(C)) such
that xr = Pm + fr, Mr(A(C)) and
xrl = sup{llp(X) : X e IC, X is A nilpotent}.
For w c Fd, define B(U) D (5rm)w = Vr.(x)wV,. Let mr denote the formal power
series Yv(xr)ww. For X KC andj = 0, 1, 2,..., it follows from From Lemma 4.1.3
that there exists 0 < p < 1 such that
II 3Y, ()w XWI < (Xm)w XWI
Iwl=j Iwl=J
< ,llXrn (5.4)
< Cpi
This implies that the series for x,(X) converges for each X e /C and moreover we have
IIXmll < IlXmll < C.
(5.5)
Recall 7 and 5() from Subsection 2.3.3. Fox 0 < t < 7 and a unit vector u c U. For
each 0 < j < and Xe IC, it follows that
c2 > It (X)w S(C)w(U 0)112
Iwl~
> tJ C (Xt)wu G wll2
Iwl j
Iwl=J
> t2 (xm)wU 2
Since f is arbitrary, letting t 1 7 implies that I(m)wll < c for all w e d and m e N.
Since U is a separable Hilbert space and the sequence {(,m)w} 1 is bounded (by
C ), for each w e Fd, there exists a subsequence of {(Im)w}L1 that converges with
respect to the WOT on B(U). By a diagonal argument similar to the one in Lemma 4.2.1,
it follows that there exists a subsequence {m,,} of {xm,} and {2w}wj c B(U) such that
for each w e Fd
(5m,)w 5w
with respect to the WOT on B(U).
Let X e IC(n) for some n. Since Hlwl(mk)w & Xw lwl w 0& X" with respect
to the WOT on B(U 0 Cn), it follows from equation (5.4) that I EY:W, XW"l < Co.
Hence the series for x(X) converges in norm.
Let e > 0 be given. Choose Lx E N such that for all k > Lx,
xj=k w= <. (5.6)
jIk wlj
Let Nx E N be such that for all k > Nx,
(5.7)
J <
j=k
Thus for all k > Nx it follows from equations (5.4) and (5.7) that,
j (k)w Xw <
j=k w=j
(5.8)
Let Mx = max{Lx, Nx}, h e U 0 Cn be a unit vector and y
From equations (5.5) and (5.8) it follows that
X")
(MxjIw
(jxw
J a W
j MI
< Iyl(C + )
Since EYM (lwl~( ),w
B(U & Cn), it follows that
x w Z" : :
2 x, Xw with respect to the WOT on
xw)
, ( r, )w
w=j
Equations (5.10) and (5.11) together imply
lyll < c +
Since h in the definition of y is arbitrary, it follows that
Mx
j=0
SIw
Iw j
XWl < c
Thus from equations (5.6) and (5.13) it follows that
X") h.
lIwl=j
5c^lmf
^1~  j^/
(5.9)
(5.10)
h, Y (y, y).
(5.11)
(5.12)
(5.13)
h, Y
S0 Mx o0
II^W(X)ll=II EE ^w XWll<II IIE^w IXWll+II Y, E^0w XWll
j o Iwl= jo Iwl=j j=Mx 1l wl=J
< C+2e.
Letting c 0 implies that I(X) < C. Since X e IC was arbitrary, it follows that
I1x < C. (5.14)
To prove the reverse inequality, observe that for w e A, (,m)w = V VmVPw VmV and
Vm V, pwV V, V pw with respect to the WOT on B(U). This implies that xw = pw for all
w e A and so by definition, we get I1x > C. Thus
Ix = C. (5.15)
[]
CHAPTER 6
INFINITE INITIAL SEGMENTS
In this chapter we present two examples of noncommutative operator domains,
and consider the CaratheodoryFej6r Interpolation problem (CFP) for these domains
under the assumption that the initial segment A is an infinite set. The examples we will
consider here will be the operator (as opposed to matrix) versions of those presented
in Subsection 2.1.1. So naturally, most of the definitions including noncommutative
neighborhood of zero, circled etc. extend analogously. A slight modification in the notion
of the 7 > 0 neighborhood of 0 is necessary. Given an operator A on a Hilbert space
H, write A > 0 if there is an c > 0 such that A > lc; i.e., A = A* and for all vectors
h e H, the inequality (Ah, h) > c(h, h) holds. In the operator version, a non commutative
7neighborhood of zero, is the set of T = (T, ..., Td) acting on H such that
d
7 2/ Y, Tj Tj*
j= 1
6.1 Examples of Noncommutative Operator Domains
Let 'H be a separable infinite dimensional Hilbert Space.
The ddimensional Noncommutative Polydisc is defined by
Cd= (T...,. Td) : T B() T 11 1<1}
Just as for the noncommutative matrix polydisc, = 1 and F = Vd.
The d x ddimensional Noncommutative Mixed Ball is defined by,
Ddd {T =(T11, T12, B Td) T E() TI o < 1}
where I TIop is the norm of the operator (Ty) : (d) ( B(d). As expected
7 = and rF = dd.
\Idd
We will demonstrate that the operator algebra approach that we used in Chapter 5
(for the finite A case) can also be applied here (to handle the infinite A case) and that it
leads to a similar necessary and sufficient condition for the solvability of the CFP.
Fix an infinite initial segment A c Td. In order to make the proofs from the Chapters
2 5 work for this setting, some minor modifications need to be made.
Recall the Noncommutative Fock Space and the dtuple of Creation Operators
S = (S1,..., Sd) from Section 2.3. A more general property of S is the following.
Let F2(A) denote the completion of the linear span of A with respect to the inner
product defined in (2.7).
Lemma 6.1.1. If V : F2(A) F2 denotes the inclusion map, then
(i) (V*SV)" = V*SV for all w Fd.
(ii) V*SV is Anilpotent, i.e. (V*SV)" = 0 for all w i A.
Here V*SV = (V*51V,..., V*SdV).
Proof. We prove item (i) by induction. When Iwl = 0 or 1, the statement is true. Assume
the statement is true for all words of length at most n. Let w be a word of length n + 1.
Then w = vwg for some word w of length n and some j such that 1 < j < d. Let u e A.
(V*SV)w(u) = (V*sV)w(V*SV)s(u)
= (V*SWV)(V*S V)(u)
SV*S(gju) if gju E A
0 if gu A
Swu if wu A
0 if wu A
=(V*SWV)(u).
Thus we get (V*SV) = V*SwV for all w such that Iwl = n + 1, and this completes
the proof.
To prove item (ii), fix w ( A. In lieu of Part (i), it suffices to show that V*SwV = 0.
Let u e A. Since w A and A is an initial segment, we get wu A. It follows that
V*SwV(u) = V*(wu) = 0.
D
6.2 The ddimensional Noncommutative Polydisc
In this section we pose the CFP for Cd, the ddimensional Noncommutative
Polydisc, with the infinite initial segment A c Fd and give our main result. We begin with
the following remark. Let V be as in Lemma 6.1.1.
Remark 6.2.1. If 0 < t < 1, then
(i) tS = (tS1,..., tSd) Cd
(ii) tV*SV e Cd.
We define A(Cd) and Z(Cd) as before (See Sections 4.1 and 4.5). The proofs
from Sections 4.1, 4.2 and 4.3, can be generalized to this current setting (A is infinite) by
replacing S(f) by S.
In order to generalize the proof of Lemma 4.5.1, to the current setting, we replace
S() with V*SV. For clarity, we present the modified argument here.
Let 0 < t < 1 be given. Since fm e Mq(Z(Cd)), using Lemma 6.1.1 (ii), it follows that
fm(tV*SV) = 0. Since fm f in norm, we have f(tV*SV) = 0. If x e Cq is a unit vector,
then
0 = If(tV*SV)12
> IIf(tV*SV)(x 08)12
= ti IY fx l12
j=0 Iwl=j,w6A
> t21wl11 fxll2
for all w e A. Hence f = 0 for all w e A.
We now state the CFP and our main result.
Let U be an infinite dimensional separable Hilbert space, A c Fd be an infinite initial
segment and p = EwA pwW be a formal power series such that pw e B(L) and
pl = sup{p(T) : T C Cd < .
Does there exist x = Cv,,e ww such that x2 = pw for all w c A and 112ll < 1?
Theorem 6.2.1. There exists a solution x to the above problem if and only if
sup{p(T)I : T Cd, T is A nilpotent} < 1.
As before, we will prove Theorem 6.2.1 by first proving it for matrix coefficients, i.e.
pw e Mq and then by following that with a WOT approximation argument.
Let 7 : A(Cd)/00(Cd) B(MA) denote a completely isometric algebra
homomorphism obtained by applying Theorem 3.2.1 to the abstract unital operator
algebra A(Cd)/Z(Cd). For 1 < j < d, let R = r(gj +Z(Cd)), and R = (R1, R,,..., Rd).
Since Ig + (Cd)ll < I 1 and 7 is isometric, we have IIRjI < 1. Moreover R is
Anilpotent.
In Section 5.1, p was a polynomial, which automatically gave us the finiteness
of p(R)II. But here, since A is infinite, we will need an approximation argument to
generalize Proposition 5.1. We present the generalization below.
Proposition 6.1. There exists f e Mq(I(Cd)) such that lip + f l = lip + M(I(Cd))l =
sup{lp(T) : T Cd, T is A nilpotent}.
Proof. We know from the generalization of Proposition 4.3 to the current setting of
infinite A, that there exists an f e Mq(I(Cd)) such that
lp+ fl = lp+ Mq,((Cd))l.
Let 0 < t < 1. Define the formal power series pt by
Pt = tlPww.
wEA
Since for each T c Cd we have pt(T) = p(tT), it follows that lptll < lpll. It is also
true that if 0 < t1 < t2 < 1, then IlptllI< Ilpt21.
For 0 < t < 1, define p = wEA,.Iwlk tlwpww and pk = wEA,.wl
denote the map Iq & : Mq(A(Cd)d)/Mq(Z(Cd)) M, B(A4).
We will first prove that I,(pt + Mq ((Cd))l = Ipt(R)II = IIp(tR)ll.
We know that
(p + MQ(Z(Cd)) = pk(R) = pk(tR)
for each k. Moreover, for each T E Cd and each j, we have
II Pw Twl < Ilpll.
Iwlj,wEA
Let e > 0 be given. Choose a natural number N such that
t < lipi
j= N+ 1
Let T e Cd be arbitrary. For k > N we have,
( p )(T)II = 1 t P., 0 T II
j= k 1 wjwEA
< till Pw & TWII
j k 1 Iwj,wEA
00
< HIpil Y tj < C.
This implies that ip P ptll < c for all k > N; i.e. the sequence of partial sums of Pt
converges to pt in norm.
Since (p Mq(I(Cd))) (Pt + Mq((Cd)))ll I pPt we get that for all k > N,
(p M(I(Cd))) (Pt + M((C)))d < .
Since 7q is continuous (it is an isometry), we have
Pt (R) = (p + Mq(I(Cd))) 7r,(pt + MZq(dC)).
Thus ~q(Pt + Mq((Cd))) = pt(R) which in turn implies that
q(Pt + Mq(I(Cd))) = IPt(R)II. (6.1)
Let {t,,} be an increasing positive sequence that converges to 1. Choose h, e
Mq(I(Cd)) such that
Ipt, + hm,= IIPt, + Mq((Cd)) 1 (6.2)
for each m.
We have
IPt, hmll = Ipt + Mq(Z(Cd))ll < Ptjl < IIP
It follows that the sequence {h,} is bounded (by 2 lp). Therefore, by the weak
compactness property of the algebra A(Cd) we get a subsequence of {h,} which we will
again denote by {h,} and g e Mq(I(Cd)) such that
h( T) g(T)
for each T e Cd.
Moreover, since pt (T) p(T) it follows that
II(pt + hm)(T) i (p + g)(T)II
for each T e Cd.
Given e > 0, choose H e Cd such that
lp + gl < I(p g)(H) + (6.3)
Choose a natural number N such that for all m > N we have
I(p g)(H) < (t, + hm)(H)ll + (6.4)
From equations (6.3) and (6.4), we have,
ip + g (l < I(pt, + hm)(H)I + < IPt, + hmll + (6.5)
for all m > N. On the other hand,
lip + g>l > IIpt + g9t, > It, + hml > IIP + hm c (6.6)
for all m > N, Combining equations (6.5) and (6.6), we get
I Ipt, + hmll p + 9gl I <
for all m > N. i.e. pt + hml liP + gll.
Combining equations (6.1) and (6.2), and using the fact that 7q is isometric and
yields,
Ipt, + hmll = II(tmR)ll.
Thus IIp(tmR)II lip + g. Let C = sup{p(T) : T E Cd, T is Anilpotent}. Since
Ip(tmR) < C, it follows that
lip + gl C.
Hence,
llp+ fll = lp+ Mq((Cd)) < p+ gl
On the other hand, by definition,
ip + fl > sup{(p + f)(T) : T E Cd, T is Anilpotent} = C
and this completes the proof.
The matrix version of Theorem 6.2.1 follows as a consequence by setting x = p + f
in the above proposition.
To pass from the case of matrix coefficients to the case of operator coefficients, i.e.
to prove Theorem 6.2.1, we can imitate the proof of Theorem 5.3.1 by using the dtuple
S in place of 5(.).
6.3 The d x d Noncommutative Mixed Ball
In this section we pose the CFP for Ddd, the d x d Noncommutative Mixed Ball, with
the infinite initial segment A and give our main result.
Let Fd be the semigroup generated by the dd symbols {g}j 1. Let F2 denote the
corresponding Noncommutative Fock Space and S = (51, S12,..., Sd), the ddtuple of
Creation Operators. Fix an infinite initial segment A c Fda.
As in Section 6.2, we begin with the following remark. Let V : F2(A) IF2 be the
inclusion map.
Remark 6.3.1. If 0 < t < , then
,dd
(i) tS= (tS11, 512, .. tSd. ) Edd
(ii) tV*SV e d.
We define A(Ddd)a and _(VDdd) as before (See Sections 4.1 and 4.5). The proofs
from Sections 4.1, 4.2 and 4.3, can be generalized to this current setting (A is infinite) by
replacing 5(f) by the ddtuple S.
In order to generalize the proof of Lemma 4.5.1, to the current setting, we replace
S() with the ddtuple V*SV, and modify the proof as we did in Section 6.2.
We now state the CFP and give our main result.
Let U be an infinite dimensional separable Hilbert space, A c Fd be an infinite
initial segment and p = Y^pw, w be a formal power series such that pw e B(U) and
lp = sup{lp(T) : T EC dd} < o.
Does there exist x = Y,,d xRw such that x2 = pw for all w e A and 11 < 1?
Theorem 6.3.1. There exists a solution x to the above problem if and only if
sup{p(T) : T cDdd, T is A nilpotent} < 1.
The same strategy that we used to prove Theorem 6.2.1 can be used to Theorem
6.3.1 as well.
As before, let 6.1, let 7 : A(Ddd)/I(dd) B(M) denote a completely isometric
homomorphism obtained by applying Theorem 3.2.1 to the abstract unital operator
algebra A(Ddd)/Z00 (d). And for 1 < i < d and 1
R = (R11, R12, .., Rd). It follows that R is Anilpotent.
To prove the matrix version of Theorem 6.3.1, we can imitate the proof of Proposition
6.1. The only point that needs clarification is that the ddtuple tR for 0 < t < 1 lies in
Dda, i.e. ItRop < 1.
To see this we first observe that the formal power series
d, d
E EgU E M1(A(DddY))
i,j= 1
has norm at most one, where = max{d, d}, and Ey is the x matrix whose (i,j)th
entry is 1 and other entries are 0; 1 < i < d, 1 < j < d.
The fact that the map F : M P(A(Ddda))/M(Z(Pdd)) Me BA(M) is isometric
implies that
d,d
d,d
RE= g E,, M (
i,j= 1
ij=i
d, d
= Eg+ M(I()Ddd))
d,d
ij= 1
< 1.
To pass from the matrix version to the operator version, i.e. to prove Theorem 6.3.1,
we can imitate the techniques in the proof of Theorem 5.3.1 by using the ddtuple S in
place of 5(f).
CHAPTER 7
FUTURE RESEARCH
The following questions came up while this work was in progress: Could the
operator algebras approach used in the examples discussed in Chapter 6 be generalized
to handle noncommutative domains that are defined by a possibly infinite collection of
Linear Matrix Inequalities? Is the algebra A(IC)" a dual algebra?
It would be interesting to know the answers to these questions.
REFERENCES
[1] J. Agler, On the representation of certain holomorphic functions defined on a
polydisc, Topics in operator theory: Ernst D. Hellinger memorial volume, volume 48
of Oper. Theory Adv. Appl., pp 4766. Birkhauser, Basel, 1990.
[2] J. Agler, J.E. McCarthy, Pick Interpolation and Hilbert Function Spaces, AMS
Publication, 2002.
[3] J.A. Ball, V. Bolotnikov, Interpolation in the noncommutative SchurAgler class, J.
Operator Theory 58 (2007), no. 1, 83126.
[4] J.A. Ball, G. Groenewald, T. Malakorn, Conservative structured noncommutative
multidimensional linear systems, The state space method generalizations and
applications, 179223, Oper. Theory Adv. Appl., 161, Birkhuser, Basel, 2006.
[5] J.A. Ball, D. KalyuzhnyiVerbovetzkii, Conservative dilations of dissipative
multidimensional systems: the commutative and noncommutative settings.
Multidimens. Syst. Signal Process. 19 (2008), no. 1, 79122.
[6] J.A. Ball, W.S. Li, D. Timotin, T T Trent, A commutant liftint theorem on the
polydisc, Indiana Univ. Math. J., 48(2):653675, 1999.
[7] J. Cuntz, Simple C*algebras generated by isometries, Comm. Math. Phys. 57,
173185 (1977).
[8] C. Caratheodory, L. Fejer, Uber den Zusammenhang der Extremen von
harmonischen Funktionen mit ihren Koeffizienten und uber den PicardLandauschen
Satz, Rend. Circ. Mat. Palermo, 32: pp 218239, 1911.
[9] T. Constantinescu, J. L. Johnson, A note on noncommutative interpolation. Canad.
Math. Bull., 46(1):5970, 2003.
[10] J.B. Conway, A Course in Functional Analysis, SpringerVerlag Publications, 1985
[11] Sh.A. Dautov, G. Khudaiberganov, The CaratheodoryFej6r problem in
higherdimensional complex analysis, Sibirsk. Mat. Zh. 23 (2) (1982) 58.64, 215.
[12] M. De Oliviera, J.W. Helton, S. McCullough, M. Putinar, Engineering Systems and
Free Real Algebraic Geometry, Emerging Applications of Algebraic Geometry, IMA
Vol. Math. Appl. 149 (2009) 47 87.
[13] E.G. Effros, S. Winkler, Matrix convexity: operator analogues of the bipolar and
HahnBanach theorems, J. Funct. Anal. 144 (1997), no. 1, 117152.
[14] J. Eschmeier, L. Patton, M. Putinar, CaratheodoryFejer interpolation on polydisks,
Math. Res. Lett. 7 (2000), no. 1, 2534.
[15] C. Foias, A.E. Frazho, The commutant lifting approach to interpolation problems,
Operator Theory: Advances and Applications, vol. 44, Birkhuser, Verlag, Basel,
1990.
[16] J.W. Helton, S. McCullough, Every free basic semialgebraic set has an LMI
representation, arXiv:0908.4352v2
[17] J.W. Helton, S. McCullough, V. Vinnikov, Noncommutative convexity arises from
Linear Matrix Inequalities, J. Funct. Anal 240 (2006), no. 1, pp 105 191.
[18] J.W. Helton; S.A. McCullough; M. Putinar; V. Vinnikov: Convex Matrix Inequalities
Versus Linear Matrix Inequalities, IEEE Trans. Automat. Control 54 (2009) 952964.
[19] J.W. Helton, I. Klep, S. McCullough, N. Slinglend, Noncommutative ball maps, J.
Funct. Anal. 257 (2009), no. 1, 4787.
[20] D. KalyuzhnyiVerbovetzkii: Caratheodory Interpolation on the Noncommutative
Polydisk, J. Funct. Anal., 229 (2005), pp. 241276.
[21] D. KalyuzhnyiVerbovetzkii, V. Vinnikov, Foundations of noncommutative function
theory, in preparation.
[22] R.A. MartinezAvendano, P. Rosenthal, An Introduction to Operators on the
HardyHilbert Space, Springer Publications, 2007.
[23] V. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge
University Press, 1st edition, Jan 2003.
[24] G. Popescu, Free holomorphic functions on the unit ball of B(H)n, J. Funct. Anal.
241 (2006), pp 268333.
[25] G. Popescu, Free holomorphic functions and interpolation, Math. Ann. 342 (2008)
130.
[26] G. Popescu, Interpolation problems in several variables, J. Math. Anal. Appl.,
227(1 ):227250, 1998.
[27] W. Rudin, Principles of Mathematical Analysis, McGrawHill
Science/Engineering/Math; 3rd edition, Jan 1976.
[28] M. Rosenblum, J. Rovnyak, Hardy Classes and Operator Theory, Dover
Publications Inc., New Ed edition, Jun 1997.
[29] D. Sarason, Generalized interpolation in H", Trans. Amer. Math. Soc. 127 (1967),
pp. 179203.
[30] I. Schur, Uber Potenzreihen die im Innern des E inheitskreises beschrankt sind, J.
Reine Angew. Math., 147: pp 205232, 1917.
[31] 0. Toeplitz, ber die Fouriersche Entwickelung positive Funktionen, Rend. Circ. Mat.
Palermo 32 (1911) 191192.
[32] D. V. Voiculescu, Free Probability Theory, American Mathematical Society, 1997.
[33] D. V. Voiculescu, Free analysis questions I: Duality transform for the coalgebra of
X : B, Int. Math. Res. Not. 16 (2004) 793.822.
[34] D. V. Voiculescu, K. J. Dykema, A. Nica, Free Random Variables: a
noncommutative probability approach to free products with applications to random
matrices, operator algebras, and harmonic analysis on free groups, American
Mathematical Society, 1992.
BIOGRAPHICAL SKETCH
Sriram Balasubramanian was born in Chennai, India. He obtained his bachelor's
degree from the University of Madras and a master's degree in mathematics from the
Indian Institute of Technology Madras, before coming to the University of Florida for
doctoral study. His interests other than mathematics are the game of Cricket and South
Indian Classical and Film Music.
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Firstandforemost,IwouldliketoexpressmydeepestgratitudetomyadvisorDr.ScottMcCullough,forhiseverpresentguidance,encouragementandhelp,withoutwhichthisdissertationwouldneverhavebeencompleted.IamforevergratefultohimforhavingmeashisstudentandforinspiringmewithhistremendousknowledgeandexpertiseinMathematics.IwouldalsoliketothankDr.JamesBrooks,Dr.JamesHobert,Dr.MichaelJuryandDr.LichienShenforservingonmydoctoralcommittee.MyspecialthankstoDr.MichaelJuryforhismanyvaluablesuggestions.Finally,IwouldliketothankmyfamilyandmyfriendsespeciallyBalaji,Krishna,SubbuandVijay,fortheirconstantencouragementandsupport. 4
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page ACKNOWLEDGMENTS .................................. 4 ABSTRACT ......................................... 7 CHAPTER 1INTRODUCTION ................................... 8 1.1SummaryofResults .............................. 14 1.2Organization .................................. 16 2MATRIXCONVEXITY ................................ 18 2.1MatrixConvexSetsinCd 18 2.1.1Examples ................................ 19 2.1.2Properties ................................ 19 2.2MatricialHahnBanachSeparation ...................... 21 2.3TheNoncommutativeFockSpaceandCreationOperators ........ 26 2.3.1TheFreeSemigroupondLettersandIntialSegments ....... 26 2.3.2TheNoncommutativeFockSpace .................. 27 2.3.3TheCreationOperators ........................ 27 3ABSTRACTOPERATORALGEBRAS ....................... 29 3.1AbstractOperatorAlgebra ........................... 29 3.1.1Examples ................................ 31 3.1.2TheQuotientOperatorAlgebra .................... 31 3.2RepresentationsofAbstractUnitalOperatorAlgebras ........... 34 4THEABSTRACTOPERATORALGEBRASA(K)1&A(K)1=I(K) 35 4.1TheAlgebraA(K)1ofScalarFormalPowerSeries ............. 35 4.1.1FormalPowerSeries .......................... 35 4.1.2TheVectorSpaceA(K)1 36 4.1.3MatrixNormsonA(K)1 37 4.1.4TheAlgebraA(K)1 39 4.2WeakCompactnessandA(K)1 44 4.3TheAbstractOperatorAlgebraA(K)1 47 4.4CompletelyContractiveRepresentationsofA(K)1 48 4.5TheAbstractOperatorAlgebraA(K)1=I(K) 53 4.6AttainmentofNormsofClassesinMq(A(K)1)=Mq(I(K)) 54 5THENONCOMMUTATIVECARATHEODORYFEJERPROBLEM ....... 56 5.1TheCaratheodoryFejerInterpolationProblem(CFP) ............ 56 5.2TheMatrixVersion ............................... 56 5
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............................. 58 6INFINITEINITIALSEGMENTS ........................... 62 6.1ExamplesofNoncommutativeOperatorDomains ............. 62 6.2TheddimensionalNoncommutativePolydisc ............... 64 6.3Thed~dNoncommutativeMixedBall ................... 69 7FUTURERESEARCH ................................ 72 REFERENCES ....................................... 73 BIOGRAPHICALSKETCH ................................ 76 6
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WeposetheCaratheodoryFejerinterpolationproblemforopen,circledandboundedmatrixconvexsetsinCd.ByusingtheBlecherRuanSinclaircharacterizationofanabstractunitaloperatoralgebra,weobtainanecessaryandsufcientconditionfortheexistenceofaminimumnormsolutiontotheproblem. 7
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AclassicalinterpolationprobleminfunctiontheoryistheCaratheodoryFejerinterpolationproblem(CFP):Givenn+1complexnumbersc0,c1,...,cndoesthereexistacomplexvaluedanalyticfunctionf(z)=P1j=0fjzjdenedontheopenunitdiscDCsuchthatfj=cjforall0jnandjf(z)j1forallz2D? TheproblemandsomeofitsvariantswerestudiedbyCaratheodory,FejerandSchurduringtheearly20thcenturyin[ 30 ],[ 31 ]and[ 8 ].AnecessaryandsufcientconditionforthesolvabilityoftheCFP,whichiscommonlyreferredtoastheSchurCriterion,isthattheToeplitzmatrix isacontraction. AnoperatortheoreticviewoftheCFPwasrstpresentedbySarasoninhispioneeringwork[ 29 ].HisformulationhashadamajorimpactnotonlyontheCFPandtherelatedPickinterpolationproblembutthedevelopmentofoperatortheoryandthestudyofnonselfadjointoperatoralgebrasgenerally.WenowpresentaproofoftheequivalencebetweenthesolvabilityoftheCFPandtheSchurCriterionthatusesSarason'sideas.Wewillbeginwithsomedenitionsandstatesomewellknownfacts(withoutproofs). LetH2(D)denotetheHardyHilbertSpacedenedbyH2(D)=(f:D!C:f(z)=1Xn=0anznand1Xn=0janj2<1).
PAGE 9
LetL2denotetheHilbertspaceofsquareintegrablefunctionsonTwithrespecttothenormalizedLebesguemeasure.Theinnerproductisdenedas:hf,gi=1 2Z20f(ei) Asisoftendone,wewillviewL2asaspaceoffunctionsratherthanasaspaceofequivalenceclassesoffunctions,byidentifyingtwofunctionstobeequaliftheyareequala.e.withrespecttothenormalizedLebesguemeasure.i.e.L2=f:T!C:1 2Z20jf(ei)j2d<1. 7 ].Theexpansionofthefunctionf2L2withrespecttothisorthonormalbasisiscalledtheFourierSeriesexpansionoff.Wewritef(ei)1Xn=bf(n)ein 2Z20f(ei)eind 9
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22 ])thatforf2H2(D),theradiallimit~f(ei)=limr!1f(rei)existsforalmostall.Wewillcall~ftheboundaryfunctionoff.Animportantpropertyof~fisthat~f2L2.Thelinearmappingthattakesf2H2(D)to~f2L2isanisometryfromH2(D)ontoaclosedsubspaceofL2whichwewilldenotebyH2.Ifg2L2isgivenbyg(ei)1Xn=0bg(n)ein,thenitiswellknownthatthefunctionfdenedontheunitdiscbyf(z)=1Xn=0bg(n)znisinH2(D)andinaddition,~f=g.Conversely,iff2H2(D)isgivenbyf(z)=1Xn=0fnznthen,theFourierseriesexpansionof~fis~f(ei)1Xn=0fnein.ThisequivalencegivesusthefollowingwaytoviewthespaceH2intermsofFouriercoefcients,namely,H2=fg2L2:bg(n)=0forn<0g.
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(i) (iii) (iv) IfTS=ST,thenthereexistsafunction2H1suchthatT=PTjKwhereTistheanalyticToeplitzoperatorwithsymbolandkk1=kTk. 29 ]. 1.1 )hasnormatmostone. Proof. 1.0.1 andequation( 1.2 )that~f2H1and,1supfjf(z)j:z2Dgk~fk1=kT~fk.
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1.1 )isthecompressionoftheanalyticToeplitzoperatorT~ftothesubspacespannedbytheorthonormalseth1,z,z2,...,zniofH2(andcomputedwithrespecttothisbasis),itfollowsthatthenormofthatmatrixisatmost1. Fortheconverse,weapplyLemma 1.0.1 with(z)=zn+1,K=h1,z,z2,...,zniandT:K!Kbeingthematrixin( 1.1 ).SupposethatkTk1.LetSandPbetheoperatorsthatappearinthehypothesisofLemma 1.0.1 .ThematrixofSwithrespecttothebasisf1,z,z2,...,zngisgivenby ThusTS=STandLemma 1.0.1 impliesthatthereexists2H1suchthatT=PTjKandkk1=kTk. 1.0.1 thatsupfjf(z)j:z2Dgkk1. Fromtheoperatortheory/algebrapointofview,theCFPisessentiallyunchangedifthecoefcientsc0,c1,...,cnaretakentobeelementsofB(U)forsomeseparableHilbertspaceU.Indeed,eveninthiscase,anecessaryandsufcientconditionforthesolvabilityoftheCFPisthesameasbefore,onlynowtheentriesoftheToeplitzmatrixin( 1.1 )areboundedoperatorsinB(U)andthematrixitselfisanelementofB(n+11U).AnalternatewayofviewingtheSchurCriterionwhichismoreconvenientforourpurposesisthefollowing. 12
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Proof. 1.3 ).SinceSisacontractionandSn+1=0wehavekp(S)k1.ThefactthatthenormoftheToeplitzmatrixinequation( 1.1 )isequaltothenormoftheoperatorp(S)completestheargument. 2.DTiscalledthedefectoperatorofT.Letfejgnj=0denotethestandardorthonormalbasisofCn+1.DenetheoperatorV:H!Cn+1HbyVh=nXj=0ejDT(T)jh.
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Usingequation( 1.4 )weget,kp(T)k=knXj=0cjV(SjIH)Vk=k(IUV)(nXj=0cj(SjIH))(IUV)kknXj=0cj(SjIH)k=kp(S)k1. 1 ].Formoredetailssee[ 14 ],[ 6 ].SomeresultsontheproblemforboundedcirculardomainsinCdcanalsobefoundin[ 11 ].SomenoncommutativegeneralizationsoftheCFPhavealsobeenstudiedin[ 25 ],[ 26 ],[ 9 ],[ 20 ],[ 4 ]. Inthisthesis,someoftheexistingresultsontheCFPhavebeenextendedtothenoncommutativesettingofthefreealgebraonanitenumberofgenerators.Anexampleofadomainweconsiderhereisthed~dnoncommutativematrixmixedballdenedbyDd~d=[n2NfX=(X11,X12,...,Xd~d):XijarennmatricesandkXkop<1g.
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WhatfollowsaresomedenitionswhichwillleadustothestatementoftheCFPforthisparticulardomain. TheCFPforthed~dnoncommutativematrixmixedballisthefollowing:Let,aniteinitialsegment,andp=Xw2pww (Aspecialcaseof)Ourmainresultisthefollowing. 1.1.1 whichallowsforoperatorvaluedcoefcientspwandfw. 15
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32 ],[ 33 ],[ 34 ],[ 24 ],[ 25 ],[ 26 ],[ 21 ],[ 19 ]).Thesefunctionsarenotonlyobjectsofgreatmathematicalinterest,buthaveapplicationsinareassuchascontroltheoryandoptimization.Noncommutativepolynomials,inparticular,areofspecialinterestsincenoncommutativepolynomialinequalities(matrixinequalitieswheretheunknownsarematricestoo),occurnaturallyinthecontextofdimensionfreelinearsystems.Recentadvancesinthestudyofnoncommutativelinearmatrixinequalitiescanbefoundin[ 12 ],[ 18 ]. InChapter3,thedenitionanabstractoperatoralgebraisintroducedalongwithsomeexamples.Itisalsoshownthatthequotientofanabstractoperatoralgebrabyaclosedtwosidedidealisanabstractoperatoralgebra.ThechapterendswiththestatementoftheBlecherRuanSinclairTheoremforabstractunitaloperatoralgebras. Chapter4iswheretheinterpolatingclassA(K)1andtheidealI(K)areintroduced.ItisshownthatA(K)1andthequotientA(K)1=I(K)areabstractoperatoralgebras.SeveralkeypropertiesincludingaweakcompactnesstypepropertyofthealgebraA(K)1andthenormattainmentpropertyofthealgebraA(K)1=I(K)areestablished.ThechapterendswithadiscussiononcompletelycontractiverepresentationsofthealgebraA(K)1,whereitisshownthattuplesofnitedimensionalcompressionsofoperatorsthatgiverisetocompletelycontractiverepresentationsofA(K)1lieontheboundaryoftheunderlyingmatrixconvexset. 16
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InChapter6,aversionofTheorem 1.1.1 whereitisassumedthattheinitialsegmentisaninniteset,isprovedfortwospecialnoncommutativedomainsnamely,theddimensionalnoncommutative(operator)polydiscandthed~dnoncommutative(operator)mixedball. InChapter7,twoimportantandveryinterestingquestionsthatcameupwhilethisworkwasinprogress,areposed. 17
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Abasicobjectofstudyinthisthesisisaquantized,ornoncommutative,versionofaconvexset.Whilethedenitionseasilyextendtoconvexsubsetsofarbitraryvectorspaces,herethefocusisonsubsetsofCd,thecomplexddimensionalspace.InthischapterwepresentthedenitionofamatrixconvexsubsetofCdandintroduceourstandardassumptionsregardingthesesets.SomepropertiesandexamplesofsuchsetsandamatricialHahnBanachseparationresultarealsopresented.ThechapterendswithadiscussionoftheCreationOperatorsontheNoncommutativeFockSpace. AnoncommutativesetLisasequence(L(n))where,forn2N,L(n)Mn(Cd),whichisclosedwithrespecttodirectsums;i.e.,ifX2L(n)andY2L(m),then whereXjYj=0B@Xj00Yj1CA. AmatrixconvexsetK=(K(n))isanoncommutativesetwhichisclosedwithrespecttoconjugationbyanisometry;i.e.,if2Mm,nand=In,andifX=(X1,...,Xd)2K(m),then AsubsetUofMn(Cd)iscircledifeiUUforall2R.AmatrixconvexsetKiscircledifeachK(n)iscircled.Asacanonicalexampleofacircledmatrixconvexset, 18
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isopen; (b) isbounded;and (c) iscircled; (i) LetK(n)=f(X1,...,Xd):kXjk<1gwith=1and=p (ii) LetK(n)=fX=(X11,X12,...,Xd~d):kXkop<1g,wherekXkopisthenormoftheoperatorX=(Xij)d,~di,j=1:(Cn)~d!(Cn)d,with=1 (iii) LetK(n)=f(X1,...,Xd):supfPdj=1jhXjy,yij:kyk=1g<1gwith=1 (i) (ii) If02L,thenLisclosedwithrespecttoconjugationbyacontraction;i.e.inequation( 2.2 )canbeassumedtobeacontraction. (ii) TheclosureofLnamely 19
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2. isunitary.Therefore,~~X~=Im+n0m+nJ(~)bXJ(~)0B@Im+n0m+n1CA2K(m+n).
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LetX=(X1,...,Xd)2 L(p)andY=(Y1,...,Yd)2 L(q).ChoosesequencesXm=(Xm1,...,Xmd)andYm=(Ym1,...,Ymd)fromL(p)andL(q)respectivelysuchthatXm!XandYm!Y. Wehave,kj(XmYm)(XY)kj=dXj=1k(XmjYmj)(XjYj)k=dXj=1k(XmjXj)(YmjYj)k=dXj=1maxfkXmjXjk,kYmjYjkg L(p+q). If2Mp,`isanisometry,thenwehavekjXmXkj=dXj=1k(XmjXj)kdXj=1kXmjXjk. L(`). 13 ]).Thefollowingcontents,areminorvariantsoflemmasandtheoremsfrom[ 16 ]. Givenapositiveintegern,letTndenotethecollectionofallpositivesemidenitenncomplexmatricesoftraceone.AT2TncorrespondstoastateonMn,viathe 21
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Proof.
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Choosing~T=whereisaunitvectorsuchthatkCk=kCk,
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2.1.2 ,CYC2C(n)andsobythehypothesisofthelemma,therighthandsideoftheaboveequationisnonnegative.IfCdoesnothavenorm1,butisnotzero,thenasimplescalingargumentshowsthatfY,C(~T)0. HencebyLemma 2.2.1 ,thereexistsanS2TnsuchthatfY,C(S)0foreverym,Y2C(m)andC2Mm,n. AlinearpencilLofsizenisaformalexpressionoftheformPd`=1L`g`whereL`2Mn.ForadtupleT=(T1,...,Td)ofboundedoperatorsonaHilbertSpaceH,theevaluationofLatTisdenedastheoperatorL(T)=Pd`=1L`T`. (i) Proof. ReG(C(n))1
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IfweletR=(1)S+ ReG(CYC)tr(CRC).(2.6) Letfe1,...,edgdenotethestandardorthonormalbasisforCd.Given1`d,andcolumnvectorsc,d2Cn,deneaboundedsesquilinearformonCnbyB`(c,d)=G(R1 2 2e`) ThereexistsauniquematrixB`2MnsuchthatB`(c,d)=hB`c,di. Fixapositiveintegerm.LetY=(Y1,...,Yd)2C(m)begivenandconsiderL(Y),theevaluationofLatY.Letfe1,...,emgdenotethestandardorthonormalbasisofCm.For=Pmj=1jej2CnCm,wehavehL(Y),i=mXi,j=1dX`=1hB`j,iihY`ej,eii=mXi,j=1dX`=1G(R1 2 2e`)hY`ej,eii=mXi,j=1dX`=1G((R1 2 2 2 2 2 2
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2.6 )weget,RehL(Y),i=ReG((R1 2 2 2 2 2 2 2.5 )weget,RehL(X)nXi=1 2eiei,nXj=1 2ejeji=ReG(InXIn)>1=knXi=1 2eieik2. LetFddenotethesetofallwordsgeneratedbydsymbolsfg1,...,gdg.DenetheproductonFdbyconcatenation.i.e.,ifw=gi1...gimandv=gj1...gjn,thentheproductwvisgivenbygi1...gimgj1...gjn.Fdisasemigroupwithrespecttothisproduct,withtheemptyword;actingastheidentityelement,i.e.w;=w=;wforallw2Fd. 26
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AsetFdisaninitialsegmentifitscomplementisanidealinthesemigroupFd;i.e.,ifbothgjw,wgj2Fdn(1jd),wheneverw2Fdn.Inthecasethatd=1aninitialsegmentisthusasetoftheformf;,g1,g21,...,gm1gforsomem. ToconstructtheFockspace,F2,deneaninnerproductonChgibydening forw,v2FdandextendingitbylinearitytoallofChgi.ThecompletionofChgiinthisinnerproductisthentheHilbertspaceF2. 7 ].Given1jd,deneSj:F2!F2bySjv=gjvforawordv2FdandextendSjbylinearitytoallofChgi.ItisreadilyveriedthatSjisanisometricmappingofChgiintoitselfanditthusfollowsthatSjextendstoanisometryonallofF2.InparticularSjSj=I,theidentityonF2.Alsoofnoteistheidentity, 27
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Ofcourse,asitstandsthetupleS=(S1,...,Sd)actsontheinnitedimensionalHilbertspaceF2.Therearehowever,nitedimensionalsubspaceswhichareessentiallydeterminedbyidealsinFdandwhichareinvariantforeachSj. Thesubset(`)=fw:jwj`gofFdisacanonicalexampleofaniteinitialsegment.AndthesubspaceF(`)2ofF2spannedby(`)isinvariantforSj,1jd.LetV(`)denotetheinclusionofF(`)2intoF2andletS(`)denotetheoperatorV(`)SV(`).Thus,S(`)=((S(`))1,...,(S(`))d)where(S(`))j=V(`)SjV(`). Recallfromthedenitionofthe(open,boundedandcircled)matrixconvexsetK. Proof. 2.8 )thatP=V(`)PV(`)=V(`)dXj=1SjSj!V(`)=dXj=1(S(`))j(S(`))j.
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Webeginthischapterwiththedenitionofanabstractoperatoralgebra.Followingthatwepresentsomeexamplesandtheproofofthefactthatthequotientofanabstractoperatoralgebrabyaclosedtwosidedidealisanabstractoperatoralgebra.Furthermore,wepresentacharacterizationofanabstractunitaloperatoralgebraduetoBlecher,RuanandSinclair. AmatrixnormedspaceVissaidtobeanabstractoperatorspaceifkXYkp+`,q+r=maxfkXkp,q,kYk`,rg Itisimportanttonotethat,withoutlossofgenerality,wecanreplacetherectangularmatricesintheabovedenitionswithsquarematrices. Beforewelookatexamplesofabstractoperatoralgebras,wepresent,asaremark,aninterestingfactabouttheabstractoperatorspaceCd. 29
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Proof. SinceL(n)isconvex,closed,bounded,absorbingandcircled,itistheclosedunitballofMn(Cd)withrespecttosomenorm,whichwewilldenotekkn.WeneedtoshowthatCdtogetherwiththesequenceofnormskknisanabstractoperatorspace. LetX2Mn(Cd)besuchthatkXkn=1andA,B2Mnbeofunitnorm.ConsidertheJuliamatrices(see( 2.4 ))J(A),J(B)ofAandB.SinceJ(A)andJ(B)areunitary,itfollowsfromequation( 3.1 )that,kJ(A)(X0)J(B)k2n1. 30
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Since(Cd,kkn)isamatrixnormedspaceitfollowsthat,kXYkn+mmaxfkXkn,kYkmg. LetHbeaHilbertspaceandV=B(H).DenematrixnormsonVbyk(Tij)kp,q=kTk (ii) LetHbeanarbitraryseparableHilbertspace,andVdenotethealgebraofpolynomialsindvariables.DenematrixnormsonVbyk(xi,j)kp,q=supfk(xi,j(T))kg
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Proof. Observethat Usingequations( 3.2 )and( 3.3 )gives,kjAXBkjp,q=kj(A(xij+yij)B)kjp,qkA(xij+yij)Bkp,qkAkk(xij+yij)kp,qkBk
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3.4 ),( 3.5 )and( 3.6 )yields, Proof. 3.1.2 weknowthatV=Wisanabstractoperatorspace.Moreover,1+WistheunitofV=W.ItremainstoshowthattheproductonV=Wiscompletelycontractive.Forthatpurpose,letX=((xij))andY=((yij))2Mp(V=W).Choose(aij),(bij)2Mp(W)suchthatkj((xij))kjp+>k(xij+aij)kp SinceWisatwosidedidealofV,itfollowsthat 33
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3.7 ),( 3.8 ),( 3.9 )andthefactthatmultiplicationinViscompletelycontractiveyields,kjXYkjp=kj((xij+aij)(yij+bij))kjpk(xij+aij)(yij+bij)kpk(xij+aij)kpk(yij+bij)kp<(kj((xij))kjp+)(kj((yij))kjp+)=(kjXkjp+)(kjYkjp+). LetVandWbeabstractoperatorspacesand:V!Wbealinearmap.Deneq:MqV!MqWbyq=Iq,whereIqistheqqidentitymatrix. Themapissaidtobecompletelycontractive(isometric)ifqisacontraction(isometry)foreachq2N. Acompletelycontractive(isometric)representationofanalgebraAisacompletelycontractive(isometric)algebrahomomorphism:A!B(M)forsomeHilbertspaceM.
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2.1.1 .Inparticular,K(1)isancircledopenconvexsubsetofCd.Thischapterisdividedintotwoparts.Intherstpart,whichconsistsoffoursubsections,weconstructanabstractunitaloperatoralgebrawhichisanaturalnoncommutativeanalogoftheBanachalgebraH1(K(1)).Wealsopresentafewlemmasoncompletelycontractiverepresentationsofthisalgebra. Inthesecondpart,Sections 4.5 and 4.6 ,weconsidertheidealI(K)ofthealgebraA(K)1determinedbyaniteinitalsegment.WeshowthatthequotientalgebraA(K)1=I(K)determinedbytheidealisanabstractunitaloperatoralgebra.Wealsoshowthatnormsofclassesinthequotientalgebraareattained. wherefw2B(U,U0).Itisconvenienttosumfaccordingtoitshomogeneousofdegreejterms;i.e., Recall,foradtupleT=(T1,...,Td)ofoperatorsonacommonseparableHilbertspaceHandawordw=gi1gi2...gin2Fd,i1,...,in2f1,2,...,dg,theevaluationofwatT
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providedthesumconvergesintheoperatornorminB(UH,U0H)intheindicatedorder.WenoteforclaritythatifU=U0=C,thenin( 4.3 )istheusualscalarproductandiff(T)converges,itisanelementofB(H). RecallthematrixconvexsetK=(K(n))whichsatisestheconditionsofAssumption 2.1.1 .WewillwriteX2KtodenoteX2Sn2NK(n).Fortheformalpowerseriesfasabove,wedene 4.4 )canbeinnite.Weareonlyinterestedinthoseformalpowerseriesfforwhichthisisnotthecase.LetA(K)1=(f=Xw2Fdfww:fw2C,kfk<1). Proof. 2.3.3 .For 36
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2.3.1 andRemark 2.3.1 ,weget0=kfk2 Thusfw=0forallwsuchthatjwj`.Since`isarbitrary,thelemmafollows. ThefollowingLemmasplaysanimportantroleintheanalysistofollowgenerally,andinprovingthatA(K)1isanalgebra,inparticular. Proof. 2.3 )andS(`)fromSubsection 2.3.3 .Fix0t
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Letfe1,e2....,eqgdenotethestandardorthonormalbasisforCq.Weknowthatforeachw2Fd, 2.(4.11) Moreover,foreach1iq,equation( 4.10 )impliesthat Usingequations( 4.11 )and( 4.12 ),weget1Xj=02jXjwj=jkfwk21Xj=02jXjwj=j(qXi=1kfweik2)=qXi=1(1Xj=02jXjwj=jkfweik2)qkfk2.
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Proof. 2Z20F(eit)eijtdt. 2Z20kF(eit)kdt. Givenf=Pw2Fdfww2Mp,q(A(K)1)andg=Pw2Fdgww2Mq,r(A(K)1),denetheproductfgoffandgastheconvolutionproduct;i.e.,fg=Xw2FdXuv=wfugv!w.
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(i) (ii) (iii) (iv) Proof. (i)AsintheproofofLemma 4.1.3 ,letAj=Xjwj=jfwXw,Bj=Xjwj=jgwXwCj=Xjwj=j(Xuv=wfugv)Xw. LetF(z)=f(zX)andG(z)=g(zX),bothofwhicharedenedinaneighborhoodof 4.1.3 ,thereisa<1suchthatkAmkmkfkandkBkkkkgk.HencekCjk(j+1)kfkkgkj. 40
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WeclaimthattheLHSofequation( 4.13 )convergestozeroask!1.ItsufcestoshowthatthersttermontheRHSofequation( 4.13 )convergestozero,inviewoftheconvergenceofthesecond,thirdandthefourthtermsontheRHStozeroduetothefollowingreasons: (a) bothkPkm=0AmzmkandkPkn=0Bnznkarenite. (b) bothP1m=k+1AmzmandP1n=k+1Bnzn,beingtailsoftheconvergentseriesf(zX)andg(zX)respectively,convergetozeroask!1. 4.13 ). 41
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2p 2p 2p
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2r 2r 2r 20@1Xn=02nXjvj=nkgvk21A1 2r Sincetheseries1Xk=0r 4.1.2 impliesthat1Xm=02mXjuj=mkfuk2and1Xn=02nXjvj=nkgvk2arenite.Thesefactstogetherimplythedesiredconvergence.Thusfg(zX)=f(zX)g(zX)wheneverjzj<2 Fixx2CrCaandy2CpCaandconsiderthecomplexvaluedfunctionsA(z)=hFG(z)x,yiandB(z)=hF(z)G(z)x,yi.ObservethatA(z)andB(z)areanalyticonthediscfz:jzj<1 (iii)&(iv)Since,foreachX2K,fg(X)=f(X)g(X)itfollowsthatkfg(X)kkfkkgk.Thuskfgkkfkkgkandfg2Mp,r(A(K)1). 43
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Proof. Proof. 2.3.3 .Fix00,thereexistsN2Nsuchthatkfm(tS(`))fn(tS(`))k< 44
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ForeachjandX2K, FromLemma 4.1.3 ,thereisa<1suchthatforeachj, Fromequations( 4.14 )and( 4.15 ),itfollowsforeachjthat,kXjwj=jfwXwkjc, FixJ2Nsuchthat RecallfromthedenitionofthematrixconvexsetK.ChooseK2NsuchthatforallmK, foreach0jJ. 45
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4.16 )and( 4.17 ),itfollowsthatkfm(X)f(X)kJXj=0kXjwj=j((fm)wfw)Xwk+1Xj=J+1kXjwj=j((fm)wfw)XwkJXj=0Xjwj=jk(fm)wfwkj+1Xj=J+1kXjwj=j(fm)wXwk+1Xj=J+1kXjwj=jfwXwkJXj=0 (i) (ii) Thereexistsasubsequenceffmkgofffmgandfw2Mp,qsuchthat(fmk)w!fwforallw; (iii) Letf=Xw2Fdfww.ForeachX2Kthesequence(fmk(X))convergestof(X)andmoreoverkf(X)kc. Proof. 4.1.3 forX=tS(`)implythat
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Theproofofitem(ii)usesastandarddiagonalargument.Letfw1,w2,...gbeanenumerationofwordsinFdwhichrespectslength(i.e.,ifvw,thenjvjjwj).Sincek(fm)w1kc Inviewofwhathasalreadybeenproved,anapplicationofProposition 4.1 provesitem(iii). 4.9 ). Proof. 47
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4.1.4 (ii)thatforallX2K(n),kAFB(X)k=kA(X)F(X)B(X)kkAInkkF(X)kkBInkkAkkFkkBk. LetF2M`,r(A(K)1),G2Mp,q(A(K)1),X2K(n).ObservethatkFG(X)k=0B@F(X)00G(X)1CAmaxfkF(X)k,kG(X)kgmaxfkFk,kGkg. Let>0begiven.WithoutlossofgeneralityassumethatkFkkGk.Choosem2NandR2K(m)suchthatkF(R)k>kFk.Therefore Letting!0intheinequality( 4.20 )andfromtheinequality( 4.19 )itfollowsthat, Lastly,completecontractivityofmultiplicationinMp(A(K)1)followsdirectlyfromLemma 4.1.4 (iv).ThusA(K)1isanabstractoperatoralgebra. 3.2 .Theorem 3.2.1 guaranteestheexistenceofacompletelyisometricrepresentationfortheabstractunitaloperatoralgebraA(K)1. 48
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Inthissection,weprovethatforacompletelycontractiverepresentationTofA(K)1,foranyn2NandnitedimensionalsubspaceWofMofdimensionnand0t<1thetupletZ=tVTV=(tVT1V,...,tVTdV) thenfr=1Xj=0rjXjwj=jfww=1Xj=0rjfj. Proof. 4.22 ).Lemma 4.1.3 impliesthatkfjkkfk.BecauseTiscompletelycontractivekfj(T)kkfk.Itfollowsthatfr(T)convergesinnorm.Sincealsothepartialsumsoffrconverge(tofr)inthenormofMq(A(K)1),itfollowsthatT(fr)=fr(T)andsokfr(T)kkfrk. TheinequalitykfrkkfkisstraightforwardbecauserKK. NowsupposethatTiscompletelyisometric.Inthiscasekfr(T)k=kfrk.Ontheotherhandlimr!1kfrk=kfk. 49
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K. (b)IfX2K(`),thenL(X)convergesinnorm;i.e.,theseries1Xj=0L(X)j+1 (c)kL(X)k<1andhenceLisinMk(A(K)1)andhasnormatmostone. (d)IfTisacompletelycontractiverepresentationofA(K)1,then2(L(T)+L(T))0. Proof. K.Toprovetheconverse,assumethecontrary,i.e.supposethereexists~X2Kandaunitvectorvsuchthath(2L(~X)L(~X))v,vi=0.Thefollowingargumenthasbeenadaptedfrom[ 16 ].Denethemapq:R!Rbyq(t)=h(2L(t~X)L(t~X))v,vi. 50
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Toprovepart(b)ofthelemma,letX2K(`)begiven.BecauseK(`)iscircled,itfollowsthateiX2K(`)foreach.Hence, foreach.Fornotationalease,letY=L(X).ThusYisak`k`matrixandequation( 4.23 )impliesthatthespectrumofYliesstrictlywithinthedisc;i.e.,eacheigenvalueofYhasabsolutevaluelessthanone.Thus,1 21Xj=0Y Toprove(c)observethatkY(2Y)1k<1ifandonlyif(2Y)(2Y)YY Toprovepart(d),observe,SinceTiscompletelycontractiveandL2Mk(A(K)1)withnormatmostone,anapplicationofLemma 4.4.1 yields,kL(rT)k1.Arguingasintheproofofpart(b),itfollowsthat2(L(rT)+L(rT))0.Thisinequalityholdsforall0r<1andthustheconclusionofpart(c)follows. 51
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Proof. 4.4.2 .Frompart(d)ofthepreviouslemma,itfollowsthat2(L(T)+L(T))0.ApplyingIkVontheleftandIkVontherightofthisinequalitygives,2(L(Z)+L(Z))=(IkV)(2(L(T)+L(T))(IkV)0. 4.4.2 andanapplicationofTheorem 2.2.1 implythatZ2 K(n).HencetZ2K(n)forall0t<1. 4.22 )andsupposethatTisacompletelycontractiverepresentationofA(K)1intoB(M)andTisnilpotent.Thenkfr(T)ksupfkf(X)k:X2K,Xisnilpotentgforall0r<1.Moreoveriffw=0forallw62,thenkf(T)ksupfkf(X)k:X2K,Xisnilpotentg. Proof. 52
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Proposition 4.2 impliesthatrZ2K.Thus,kfr(T)k
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ByidentifyingMp,q(A(K)1=I(K))withMp,q(A(K)1)=Mp,q(I(K)),Corollary 3.1.1 impliesthatthequotientA(K)1=I(K)isanabstractoperatoralgebra.Weformallyrecordthisfact. 4.2.1 yieldsasubsequenceffmkgofffmgandf2Mq(I(K))suchthat(p+fmk)(X)!(p+f)(X) 54
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Proof. Sincek(p+fmk)(R)k!k(p+f)(R)k,thereexistsK12Nsuchthat, forallkK1.Combiningtheinequalitiesfromequations( 4.24 )and( 4.25 ),impliesthat,forallkK1, Sincekp+fmkk!kp+Mq(I(K))k,thereexistsaNaturalnumberK2suchthatforallkK2, Settingk=maxfK1,K2ginequations( 4.26 )and( 4.27 ),andletting!0yields
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InthischapterweposetheCaratheodoryFejerInterpolationproblem(CFP)forouropen,boundedandcircledmatrixconvexsetK.UsingtheresultsfromChapters24,weproveanecessaryandsufcientconditionforthesolvabilityoftheproblem. 2.1.1 .LetFdbeaniteinitialsegment,andp=Xw2pww2A(K)1 5.1.1 allowingforoperatorcoefcientsisprovedinthischapter. Thestrategyistorstprovetheresultformatrixcoefcients.ThisisdoneinSection 5.2 below.PassingfrommatrixtooperatorcoefcientsisthenaccomplishedusingwellknownfactsabouttheWeakOperatorTopology(WOT)andtheStrongOperatorTopology(SOT)onthespaceofboundedoperatorsonaseparableHilbertspace.ThedetailsareinSection 5.3 Proof. 4.5.1 and 3.2.1 itfollowsthatthereexistsaHilbertspaceMandacompletelyisometrichomomorphism:A(K)1=I(K)!B(M).As 56
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4.4 ,wewilluseRtodenotethemap. ItfollowsfromTheorem 4.3 thatthereexistsf2Mq(I(K))suchthat Thefactthatiscompletelyisometricimpliesthat SinceRisacompletelycontractiverepresentationofA(K)1,Lemma 4.4.3 impliesthat Combiningtheequations( 5.1 ),( 5.2 )and( 5.3 ),itfollowsthatkp+fksupfkp(X)k:X2K,Xisnilpotentg. (Thematrixversionof)Theorem 5.1.1 followsfromtheabovepropositionbysetting~x=p+f. 57
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Proof. Forw2,deneMm3(pm)w=VmpwVmwhereVm:Um!Uistheinclusionmap.Letpmdenotetheformalpowerseriespm=Xw2(pm)ww 5.1 ,thereexistsfm2Mm(I(K))suchthatxm=pm+fm2Mm(A(K)1)andkxmk=supfkpm(X)k:X2K,Xisnilpotentg. 4.1.3 thatthereexists0<1suchthat 58
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RecallandS(`)fromSubsection 2.3.3 .Fox0t0begiven.ChooseLX2Nsuchthatforallk>LX, 59
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Thusforallk>NXitfollowsfromequations( 5.4 )and( 5.7 )that, LetMX=maxfLX,NXg,h2UCnbeaunitvectorandy=0@MXXj=0Xjwj=j~xwXw1Ah. Fromequations( 5.5 )and( 5.8 )itfollowsthat*0@MXXj=0Xjwj=j(~xmk)wXw1Ah,y+kyk0@k~xmk(X)k+k1Xj=MX+1Xjwj=j(~xmk)wXwk1A SincePMXj=0Pjwj=j(~xmk)wXw!PMXj=0Pjwj=j~xwXwwithrespecttotheWOTonB(UCn),itfollowsthat Equations( 5.10 )and( 5.11 )togetherimply Sincehinthedenitionofyisarbitrary,itfollowsthat Thusfromequations( 5.6 )and( 5.13 )itfollowsthat 60
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Toprovethereverseinequality,observethatforw2,(~xm)w=VmVmpwVmVmandVmVmpwVmVm!pwwithrespecttotheWOTonB(U).Thisimpliesthat~xw=pwforallw2andsobydenition,wegetk~xkC.Thus 61
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Inthischapterwepresenttwoexamplesofnoncommutativeoperatordomains,andconsidertheCaratheodoryFejerInterpolationproblem(CFP)forthesedomainsundertheassumptionthattheinitialsegmentisaninniteset.Theexampleswewillconsiderherewillbetheoperator(asopposedtomatrix)versionsofthosepresentedinSubsection 2.1.1 .Sonaturally,mostofthedenitionsincludingnoncommutativeneighborhoodofzero,circledetc.extendanalogously.Aslightmodicationinthenotionofthe>0neighborhoodof0isnecessary.GivenanoperatorAonaHilbertspaceH,writeA0ifthereisan>0suchthatAI;i.e.,A=Aandforallvectorsh2H,theinequalityhAh,hihh,hiholds.Intheoperatorversion,anoncommutativeneighborhoodofzero,isthesetofT=(T1,...,Td)actingonHsuchthat2IdXj=1TjTj. TheddimensionalNoncommutativePolydiscisdenedbyCd=f(T1,...,Td):Tj2B(H)&kTjk<1g Thed~ddimensionalNoncommutativeMixedBallisdenedby,Dd~d=fT=(T11,T12,...,Td~d):Tij2B(H)&kTkop<1g 62
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5 (forthenitecase)canalsobeappliedhere(tohandletheinnitecase)andthatitleadstoasimilarnecessaryandsufcientconditionforthesolvabilityoftheCFP. FixaninniteinitialsegmentFd.InordertomaketheproofsfromtheChapters25workforthissetting,someminormodicationsneedtobemade. RecalltheNoncommutativeFockSpaceandthedtupleofCreationOperatorsS=(S1,...,Sd)fromSection 2.3 .AmoregeneralpropertyofSisthefollowing. LetF2()denotethecompletionofthelinearspanofwithrespecttotheinnerproductdenedin( 2.7 ). (i) (ii) HereVSV=(VS1V,...,VSdV). Proof.
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Toproveitem(ii),xw62.InlieuofPart(i),itsufcestoshowthatVSwV=0.Letu2.Sincew62andisaninitialsegment,wegetwu62.ItfollowsthatVSwV(u)=V(wu)=0. 6.1.1 (i) 4.1 and 4.5 ).TheproofsfromSections 4.1 4.2 and 4.3 ,canbegeneralizedtothiscurrentsetting(isinnite)byreplacingS(`)byS. InordertogeneralizetheproofofLemma 4.5.1 ,tothecurrentsetting,wereplaceS(`)withVSV.Forclarity,wepresentthemodiedargumenthere. Let0
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WenowstatetheCFPandourmainresult. LetUbeaninnitedimensionalseparableHilbertspace,Fdbeaninniteinitialsegmentandp=Pw2pwwbeaformalpowerseriessuchthatpw2B(U)andkpk=supfkp(T)k:T2Cdg<1. 6.2.1 byrstprovingitformatrixcoefcients,i.e.pw2MqandthenbyfollowingthatwithaWOTapproximationargument. Let:A(Cd)1=I(Cd)!B(M)denoteacompletelyisometricalgebrahomomorphismobtainedbyapplyingTheorem 3.2.1 totheabstractunitaloperatoralgebraA(Cd)1=I(Cd).For1jd,letRj=(gj+I(Cd)),andR=(R1,R2,...,Rd).Sincekgj+I(Cd)kkgjk1andisisometric,wehavekRjk1.MoreoverRisnilpotent. InSection 5.1 ,pwasapolynomial,whichautomaticallygaveusthenitenessofkp(R)k.Buthere,sinceisinnite,wewillneedanapproximationargumenttogeneralizeProposition 5.1 .Wepresentthegeneralizationbelow. Proof. 4.3 tothecurrentsettingofinnite,thatthereexistsanf2Mq(I(Cd))suchthatkp+fk=kp+Mq(I(Cd))k.
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For0t<1,denepkt=Pw2,jwjktjwjpwwandpk=Pw2,jwjktjwjpww.LetqdenotethemapIq:Mq(A(Cd)1)=Mq(I(Cd))!MqB(M). Wewillrstprovethatkq(pt+Mq(I(Cd))k=kpt(R)k=kp(tR)k. Weknowthatq(pkt+Mq(I(Cd))=pkt(R)=pk(tR) 66
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Letftmgbeanincreasingpositivesequencethatconvergesto1.Choosehm2Mq(I(Cd))suchthat foreachm. Wehavekptm+hmk=kptm+Mq(I(Cd))kkptmkkpk. Moreover,sinceptm(T)!p(T)itfollowsthatk(ptm+hm)(T)k!k(p+g)(T)k 67
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ChooseanaturalnumberNsuchthatforallmNwehave Fromequations( 6.3 )and( 6.4 ),wehave, forallmN.Ontheotherhand, forallmN,Combiningequations( 6.5 )and( 6.6 ),wegetjkptm+hmkkp+gkj< Combiningequations( 6.1 )and( 6.2 ),andusingthefactthatqisisometricandyields,kptm+hmk=kp(tmR)k.
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ThematrixversionofTheorem 6.2.1 followsasaconsequencebysetting~x=p+fintheaboveproposition. Topassfromthecaseofmatrixcoefcientstothecaseofoperatorcoefcients,i.e.toproveTheorem 6.2.1 ,wecanimitatetheproofofTheorem 5.3.1 byusingthedtupleSinplaceofS(`). LetFd~dbethesemigroupgeneratedbythed~dsymbolsfgijgd,~di,j=1.LetF2denotethecorrespondingNoncommutativeFockSpaceandS=(S11,S12,...,Sd~d),thed~dtupleofCreationOperators.FixaninniteinitialsegmentFd~d. AsinSection 6.2 ,webeginwiththefollowingremark.LetV:F2()!F2betheinclusionmap. (i) 4.1 and 4.5 ).TheproofsfromSections 4.1 4.2 and 4.3 ,canbegeneralizedtothiscurrentsetting(isinnite)byreplacingS(`)bythed~dtupleS. InordertogeneralizetheproofofLemma 4.5.1 ,tothecurrentsetting,wereplaceS(`)withthed~dtupleVSV,andmodifytheproofaswedidinSection 6.2 WenowstatetheCFPandgiveourmainresult. LetUbeaninnitedimensionalseparableHilbertspace,Fd~dbeaninniteinitialsegmentandp=Pw2pwwbeaformalpowerseriessuchthatpw2B(U)andkpk=supfkp(T)k:T2Dd~dg<1.
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6.2.1 canbeusedtoTheorem 6.3.1 aswell. Asbefore,let 6.1 ,let:A(Dd~d)1=I(Dd~d)!B(M)denoteacompletelyisometrichomomorphismobtainedbyapplyingTheorem 3.2.1 totheabstractunitaloperatoralgebraA(Dd~d)1=I(Dd~d).Andfor1idand1j~d,letRij=(gij+I(Dd~d)),andR=(R11,R12,...,Rd~d).ItfollowsthatRisnilpotent. ToprovethematrixversionofTheorem 6.3.1 ,wecanimitatetheproofofProposition 6.1 .Theonlypointthatneedsclaricationisthatthed~dtupletRfor0t<1liesinDd~d,i.e.ktRkop<1. Toseethiswerstobservethattheformalpowerseriesd,~dXi,j=1Eijgij2M`(A(Dd~d)1) Thefactthatthemap`:M`(A(Dd~d)1)=M`(I(Dd~d))!M`B(M)isisometricimpliesthatkRkop=kd,~dXi,j=1EijRijk=k`(d,~dXi,j=1Eijgij+M`(I(Dd~d))k
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6.3.1 ,wecanimitatethetechniquesintheproofofTheorem 5.3.1 byusingthed~dtupleSinplaceofS(`). 71
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Thefollowingquestionscameupwhilethisworkwasinprogress:CouldtheoperatoralgebrasapproachusedintheexamplesdiscussedinChapter6begeneralizedtohandlenoncommutativedomainsthataredenedbyapossiblyinnitecollectionofLinearMatrixInequalities?IsthealgebraA(K)1adualalgebra? Itwouldbeinterestingtoknowtheanswerstothesequestions. 72
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[1] J.Agler,Ontherepresentationofcertainholomorphicfunctionsdenedonapolydisc,Topicsinoperatortheory:ErnstD.Hellingermemorialvolume,volume48ofOper.TheoryAdv.Appl.,pp4766.Birkhauser,Basel,1990. [2] J.Agler,J.E.McCarthy,PickInterpolationandHilbertFunctionSpaces,AMSPublication,2002. [3] J.A.Ball,V.Bolotnikov,InterpolationinthenoncommutativeSchurAglerclass,J.OperatorTheory58(2007),no.1,83. [4] J.A.Ball,G.Groenewald,T.Malakorn,Conservativestructurednoncommutativemultidimensionallinearsystems,Thestatespacemethodgeneralizationsandapplications,179,Oper.TheoryAdv.Appl.,161,Birkhuser,Basel,2006. [5] J.A.Ball,D.KalyuzhnyiVerbovetzkii,Conservativedilationsofdissipativemultidimensionalsystems:thecommutativeandnoncommutativesettings.Multidimens.Syst.SignalProcess.19(2008),no.1,79. [6] J.A.Ball,W.S.Li,D.Timotin,T.T.Trent,Acommutantliftinttheoremonthepolydisc,IndianaUniv.Math.J.,48(2):653675,1999. [7] J.Cuntz,SimpleC*algebrasgeneratedbyisometries,Comm.Math.Phys.57,173185(1977). [8] C.Caratheodory,L.Fejer,UberdenZusammenhangderExtremenvonharmonischenFunktionenmitihrenKoefzientenunduberdenPicardLandauschenSatz,Rend.Circ.Mat.Palermo,32:pp218239,1911. [9] T.Constantinescu,J.L.Johnson,Anoteonnoncommutativeinterpolation.Canad.Math.Bull.,46(1):5970,2003. [10] J.B.Conway,ACourseinFunctionalAnalysis,SpringerVerlagPublications,1985 [11] Sh.A.Dautov,G.Khudaiberganov,TheCaratheodoryFejerprobleminhigherdimensionalcomplexanalysis,Sibirsk.Mat.Zh.23(2)(1982)58.64,215. [12] M.DeOliviera,J.W.Helton,S.McCullough,M.Putinar,EngineeringSystemsandFreeRealAlgebraicGeometry,EmergingApplicationsofAlgebraicGeometry,IMAVol.Math.Appl.149(2009)4787. [13] E.G.Effros,S.Winkler,Matrixconvexity:operatoranaloguesofthebipolarandHahnBanachtheorems,J.Funct.Anal.144(1997),no.1,117. [14] J.Eschmeier,L.Patton,M.Putinar,CaratheodoryFejerinterpolationonpolydisks,Math.Res.Lett.7(2000),no.1,25. 73
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C.Foias,A.E.Frazho,Thecommutantliftingapproachtointerpolationproblems,OperatorTheory:AdvancesandApplications,vol.44,Birkhuser,Verlag,Basel,1990. [16] J.W.Helton,S.McCullough,EveryfreebasicsemialgebraicsethasanLMIrepresentation,arXiv:0908.4352v2 [17] J.W.Helton,S.McCullough,V.Vinnikov,NoncommutativeconvexityarisesfromLinearMatrixInequalities,J.Funct.Anal240(2006),no.1,pp105191. [18] J.W.Helton;S.A.McCullough;M.Putinar;V.Vinnikov:ConvexMatrixInequalitiesVersusLinearMatrixInequalities,IEEETrans.Automat.Control54(2009)952964. [19] J.W.Helton,I.Klep,S.McCullough,N.Slinglend,Noncommutativeballmaps,J.Funct.Anal.257(2009),no.1,47. [20] D.KalyuzhnyiVerbovetzkii:CaratheodoryInterpolationontheNoncommutativePolydisk,J.Funct.Anal.,229(2005),pp.241276. [21] D.KalyuzhnyiVerbovetzkii,V.Vinnikov,Foundationsofnoncommutativefunctiontheory,inpreparation. [22] R.A.MartinezAvendano,P.Rosenthal,AnIntroductiontoOperatorsontheHardyHilbertSpace,SpringerPublications,2007. [23] V.Paulsen,CompletelyBoundedMapsandOperatorAlgebras,CambridgeUniversityPress,1stedition,Jan2003. [24] G.Popescu,FreeholomorphicfunctionsontheunitballofB(H)n,J.Funct.Anal.241(2006),pp268333. [25] G.Popescu,Freeholomorphicfunctionsandinterpolation,Math.Ann.342(2008)130. [26] G.Popescu,Interpolationproblemsinseveralvariables,J.Math.Anal.Appl.,227(1):227250,1998. [27] W.Rudin,PrinciplesofMathematicalAnalysis,McGrawHillScience/Engineering/Math;3rdedition,Jan1976. [28] M.Rosenblum,J.Rovnyak,HardyClassesandOperatorTheory,DoverPublicationsInc.,NewEdedition,Jun1997. [29] D.Sarason,GeneralizedinterpolationinH1,Trans.Amer.Math.Soc.127(1967),pp.179203. [30] I.Schur,UberPotenzreihendieimInnerndesEinheitskreisesbeschranktsind,J.ReineAngew.Math.,147:pp205232,1917. 74
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O.Toeplitz,berdieFourierscheEntwickelungpositiverFunktionen,Rend.Circ.Mat.Palermo32(1911)191192. [32] D.V.Voiculescu,FreeProbabilityTheory,AmericanMathematicalSociety,1997. [33] D.V.Voiculescu,FreeanalysisquestionsI:DualitytransformforthecoalgebraofX:B,Int.Math.Res.Not.16(2004)793.822. [34] D.V.Voiculescu,K.J.Dykema,A.Nica,FreeRandomVariables:anoncommutativeprobabilityapproachtofreeproductswithapplicationstorandommatrices,operatoralgebras,andharmonicanalysisonfreegroups,AmericanMathematicalSociety,1992. 75
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SriramBalasubramanianwasborninChennai,India.Heobtainedhisbachelor'sdegreefromtheUniversityofMadrasandamaster'sdegreeinmathematicsfromtheIndianInstituteofTechnologyMadras,beforecomingtotheUniversityofFloridafordoctoralstudy.HisinterestsotherthanmathematicsarethegameofCricketandSouthIndianClassicalandFilmMusic. 76
