Homogeneous multiplication operators on bounded symmetric domains.
type: Item
instance of: scholarly article
title: Homogeneous multiplication operators on bounded symmetric domains.
zbMATH Open document ID: 1039.47020
DOI: 10.1016/S0022-1236(02)00072-1
author: Jonathan Arazy
author: Genkai Zhang
published in: Journal of Functional Analysis
publication date: 2003-09-04T00:00:00Z
review text: The purpose of the authors is to study the spectral property of a family of multiplication tuples on the holomorphic function spaces with the reproducing kernel \(1-\langle z,w\rangle)^{-v}\) on the unit ball \(B^d\) of \(\mathbb{C}^d\), for \(0\leq v< \infty\). They also generalize this problem: Let \(D\) be an irreducible bounded symmetric domain of dimension \(d\) and let \(H_v(D)\) be the analytic continuation of the weighted Bergman spaces of holomorphic functions on \(D\). Consider the \(d\)-tuple \(M=(M_1,\dots, M_d)\) of multiplication operators by coordinate functions. The authors study the similar problem of spectral property of the \(d\)-tuple \(M\) on \(H_v(D)\). They find those parameters \(v\) for which the tuple \(M\) is subnormal and answer some open questions of \textit{B. Bagchi} and \textit{G. Misra} [ibid. 136, 171--213 (1996; Zbl 0867.47022)]. In particular, they prove that when \(D= B^d\), then \(B^d\) is a \(k\)-spectral set of \(M\) if and only if \(H_v(B^d)\) is the Hardy space or a weighted Bergman space. Also, in this case, the contractivity, hyponormality and subnormality of the multiplication tuples are investigated.
Mathematics Subject Classification ID: 47B38
zbMATH DE Number: 1974790
zbMATH Keywords: Taylor spectrum
zbMATH Keywords: \(k\)-spectral set
zbMATH Keywords: irreducible bounded symmetric domain
zbMATH Keywords: weighted Bergman space
zbMATH Keywords: multiplication tuples
zbMATH Keywords: hyponormal operator
zbMATH Keywords: subnormal operator
reviewed by: Bahmann Yousefi
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1974790