Commutants and reflexivity of multiplication tuples on vector-valued reproducing kernel Hilbert spaces

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:466

Article

Chavan, Sameer¹  Podder, Shubhankar²  Trivedi, Shailesh¹ 

[1] Indian Inst Technol Kanpur, Dept Math & Stat, Kanpur, Uttar Pradesh, India

[2] Harish Chandra Res Inst, Sch Math, HBNI, Chhatnag Rd, Allahabad 211019, Uttar Pradesh, India

关键词: Operator-valued reproducing kernel;    Multiplication tuple;    Commutant;    Reflexivity;    Weighted shift;    Directed trees;   

DOI  :  10.1016/j.jmaa.2018.06.062

来源: Elsevier

【 摘 要 】

Motivated by the theory of weighted shifts on directed trees and its multivariable counterpart, we address the question of identifying commutant and reflexivity of the multiplication d-tuple M-z on a reproducing kernel Hilbert space H of E-valued holomorphic functions on Omega, where E is a separable Hilbert space and Omega is a bounded domain in C-d admitting bounded approximation by polynomials. In case E is a finite dimensional cyclic subspace for M-z, under some natural conditions on the B(E)-valued kernel associated with H, the commutant of M-z is shown to be the algebra H-B(E)(infinity) (Omega) of bounded holomorphic B(E)-valued functions on Omega, provided M-z satisfies the matrix-valued von Neumann's inequality. This generalizes a classical result of Shields and Wallen (the case of dim E = 1 and d = 1). As an application, we determine the commutant of a Bergman shift on a leafless, locally finite, rooted directed tree I of finite branching index. As the second main result of this paper, we show that a multiplication d-tuple M-z on H satisfying the von Neumann's inequality is reflexive. This provides several new classes of examples as well as recovers special cases of various known results in one and several variables. We also exhibit a family of tri-diagonal B(C-2)-valued kernels for which the associated multiplication operators M-z are non-hyponormal reflexive operators with commutants equal to H-B(C2)(infinity). (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

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