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Entropy and Multivariable Interpolation
Gelu Popescu, University of Texas at San Antonio, TX
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Memoirs of the American Mathematical Society
2006; 83 pp; softcover
Volume: 184
ISBN-10: 0-8218-3912-8
ISBN-13: 978-0-8218-3912-6
List Price: US$60 Individual Members: US$36
Institutional Members: US\$48
Order Code: MEMO/184/868

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

"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