Electronic ISBN:  9781470404727 
Product Code:  MEMO/184/868.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 184; 2006; 83 ppMSC: Primary 47;
We define a new notion of entropy for operators on Fock spaces and positive multiToeplitz kernels on free semigroups. This is studied in connection with factorization theorems for (e.g., multiToeplitz, multianalytic, etc.) operators on Fock spaces. These results lead to entropy inequalities and entropy formulas for positive multiToeplitz kernels on free semigroups (resp. multianalytic operators) and consequences concerning the extreme points of the unit ball of the noncommutative analytic Toeplitz algebra \(F_n^\infty\).
We obtain several geometric characterizations of the central intertwining lifting, a maximal principle, and a permanence principle for the noncommutative commutant lifting theorem. Under certain natural conditions, we find explicit forms for the maximal entropy solution of this multivariable commutant lifting theorem.
All these results are used to solve maximal entropy interpolation problems in several variables. We obtain explicit forms for the maximal entropy solution (as well as its entropy) of the Sarason, CarathéodorySchur, and NevanlinnaPick type interpolation problems for the noncommutative (resp. commutative) analytic Toeplitz algebra \(F_n^\infty\) (resp. \(W_n^\infty\)) and their tensor products with \(B({\mathcal H}, {\mathcal K})\). In particular, we provide explicit forms for the maximal entropy solutions of several interpolation problems on the unit ball of \(\mathbb{C}^n\). 
Table of Contents

Chapters

Introduction

1. Operators on Fock spaces and their entropy

2. Noncommutative commutant lifting theorem: Geometric structure and maximal entropy solution

3. Maximal entropy interpolation problems in several variables


Reviews

I am compelled to say some words on how valuable this memoir is as a monograph on the subject. ...as a continuation on the author's previous papers. This memoir is a worthwhile addition.
Journal of Approximation Theory


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We define a new notion of entropy for operators on Fock spaces and positive multiToeplitz kernels on free semigroups. This is studied in connection with factorization theorems for (e.g., multiToeplitz, multianalytic, etc.) operators on Fock spaces. These results lead to entropy inequalities and entropy formulas for positive multiToeplitz kernels on free semigroups (resp. multianalytic operators) and consequences concerning the extreme points of the unit ball of the noncommutative analytic Toeplitz algebra \(F_n^\infty\).
We obtain several geometric characterizations of the central intertwining lifting, a maximal principle, and a permanence principle for the noncommutative commutant lifting theorem. Under certain natural conditions, we find explicit forms for the maximal entropy solution of this multivariable commutant lifting theorem.
All these results are used to solve maximal entropy interpolation problems in several variables. We obtain explicit forms for the maximal entropy solution (as well as its entropy) of the Sarason, CarathéodorySchur, and NevanlinnaPick type interpolation problems for the noncommutative (resp. commutative) analytic Toeplitz algebra \(F_n^\infty\) (resp. \(W_n^\infty\)) and their tensor products with \(B({\mathcal H}, {\mathcal K})\). In particular, we provide explicit forms for the maximal entropy solutions of several interpolation problems on the unit ball of \(\mathbb{C}^n\).

Chapters

Introduction

1. Operators on Fock spaces and their entropy

2. Noncommutative commutant lifting theorem: Geometric structure and maximal entropy solution

3. Maximal entropy interpolation problems in several variables

I am compelled to say some words on how valuable this memoir is as a monograph on the subject. ...as a continuation on the author's previous papers. This memoir is a worthwhile addition.
Journal of Approximation Theory