# Entropy and Multivariable Interpolation

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*Gelu Popescu*

We define a new notion of entropy for operators on
Fock spaces and positive multi-Toeplitz kernels on free semigroups. This is
studied in connection with factorization theorems for (e.g., multi-Toeplitz,
multi-analytic, etc.) operators on Fock spaces. These results lead to entropy
inequalities and entropy formulas for positive multi-Toeplitz kernels on free
semigroups (resp. multi-analytic 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, Carathéodory-Schur, and
Nevanlinna-Pick 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
\(\mathbb{C}^n\).

#### Reviews & Endorsements

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

#### Table of Contents

# Table of Contents

## Entropy and Multivariable Interpolation

- Contents v6 free
- Introduction 18 free
- Chapter 1. Operators on Fock Spaces and their Entropy 714 free
- Chapter 2. Noncommutative Commutant Lifting Theorem: Geometric Structure and Maximal Entropy Solution 2936
- 2.1. Multivariate intertwining liftings and geometric structure 2936
- 2.2. Central lifting in several variables and geometric characterizations 3744
- 2.3. A maximum principle for the noncommutative commutant lifting theorem 4350
- 2.4. A permanence principle for the central intertwining lifting 4754
- 2.5. Quasi outer spectral factorizations 4855
- 2.6. Noncommutative commutant lifting theorem and the maximal entropy solution 5461

- Chapter 3. Maximal Entropy Interpolation Problems in Several Variables 6168
- 3.1. Maximal entropy solution for the Sarason interpolation problem for analytic Toeplitz algebras 6168
- 3.2. Maximal entropy solution for the Caratheodory-Schur interpolation problem for analytic Toeplitz algebras 6774
- 3.3. Maximal entropy solution for the Nevanlinna-Pick interpolation problem with operatorial argument in several variables 6976
- 3.4. Maximal entropy interpolation on the unit ball of C[sup(n)] 7582

- Bibliography 8188