Entropy and Multivariable Interpolation (Memoirs of the American Mathematical Society) - Softcover

Gelu Popescu

Softcover

ISBN 10: 0821839128 ISBN 13: 9780821839126

Publisher: American Mathematical Society, 2006

Synopsis

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 $\mathbb{C}^n$.

Edition details

• Publisher: American Mathematical Society
• Publication date: 2006
• ISBN 10: 0821839128
• ISBN 13: 9780821839126
• Binding: Paperback
• Language: English
• Number of pages: 83