Functional calculus and multi-analytic models on regular Λ-polyballs

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:491

Article

Popescu, Gelu¹ 

[1] Univ Texas San Antonio, Dept Math, San Antonio, TX 78249 USA

关键词: Multivariable operator theory;    Lambda-polyball;    Noncommutative Hardy space;    Functional calculus;    Characteristic function;    Multi-analytic model;   

DOI  :  10.1016/j.jmaa.2020.124312

来源: Elsevier

【 摘 要 】

In a recent paper, we introduced the standard k-tuple S := (S-1, ..., S-k) of pure row isometrics S-i := [S-i,S-1 ... S-i,n(i)] acting on the Hilbert space l(2) (F-n1(+) x ... x F-nk(+)), where F-n(+) is the unital free semigroup with n generators, and showed that S is the universal k-tuple of doubly Lambda-commuting row isometries, i.e. S-i,S-s*S-j,S-t = <(lambda(ij)(s,t))over bar>S-j,(t) S-i,S-s* for every i, j is an element of {1, ..., k} with i not equal j and every s is an element of {1, ..., n(i)}, t is an element of {1, ..., n(j)}, where Lambda(ij) := [lambda(i,j)(s,t)] is an n(i) x n(j)-matrix with the entries in T := {z is an element of C : vertical bar z vertical bar = 1} and Lambda(j,i) = Lambda(i,j)*. It was also proved that the set of all k-tuples T := (T-1, ..., T-k) of row operators T-i := [T-i,T-1 ... T-i,(ni)} acting on a Hilbert space H which admit S as universal model, i.e. there is a Hilbert space D such that H is jointly co-invariant for all operators S-i,S-s circle times I-D and T-i,T-s* = (S-i,S-s* circle times I-D) vertical bar(H), i is an element of {1, ..., k} and s is an element of {1, ..., n(i)}, consists of the pure elements of a set B-Lambda (H) which was called the regular Lambda-polyball. The goal of the present paper is to introduce and study noncommutative Hardy spaces associated with the regular Lambda-polyball, to develop a functional calculus on noncommutative Hardy spaces for the completely non-coisometric (c.n.c.) k-tuples in B-Lambda (H), and to study the characteristic functions and the associated multi-analytic models for the c.n.c. elements in the regular Lambda-polyball. In addition, we show that the characteristic function is a complete unitary invariant for the class of c.n.c. k-tuples in B-Lambda (H). These results extend the corresponding classical results of Sz.-Nagy-Foias for contractions and the noncommutative versions for row contractions. In the particular case when n(1) = ... = n(k) = 1 and Lambda(ij) = 1, we obtain a functional calculus and operator model theory in terms of characteristic functions for k-tuples of contractions satisfying Brehmer condition. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free