Spectral mapping theorems for subnormal operators
type: Item
instance of: scholarly article
title: Spectral mapping theorems for subnormal operators
zbMATH Open document ID: 0538.47013
DOI: 10.1016/0022-1236(84)90083-1
published in: Journal of Functional Analysis
publication date: 1984-01-01T00:00:00Z
review text: Let S be a subnormal operator whose minimal normal extension has scalar- valued spectral measure \(\mu\). Denote the \(weak^*\) closure in \(L^{\infty}(\mu)\) of the rational functions in z with poles off \(\sigma\) (S) by \(R^{\infty}(\sigma(S),\mu)\). For \(f\in R^{\infty}(\sigma(S),\mu)\), f(S) has a natural definition and the resulting map \(f\to f(S)\) has many pleasant properties. The paper under review investigates the problem of characterizing the various closed spectra of f(S) in terms of f and the various closed spectra of S. Given \(f\in R^{\infty}(\sigma(S),\mu)\), using function theory, a natural way is found to define the image under f of \(\sigma\) (S), denoted \(f(\sigma\) (S)), and the cluster set of f on \(\sigma_ e(S)\) denoted \(c\ell(f;\sigma_ e(S))\). Notice that for K a compact superset of \(\sigma\) (S), \(R^{\infty}(K,\mu)\subseteq R^{\infty}(\sigma(S),\mu)\) where \(R^{\infty}(K,\mu)\) denotes the obvious. Call K almost open iff the set of nonpeak points for R(K) that lie off the interior of K has area zero. Call K nice iff all but countably many points of the boundary of K are points of positive lower capacity density for the complement of K. It is proved that if K is an almost open compact superset of \(\sigma\) (S), then for any \(f\in R^{\infty}(K,\mu), f(\sigma(S))\subseteq \sigma(f(S))\) and \(c\ell(f;\sigma_ e(S))\subseteq \sigma_ e(f(S)).\) It is also proved that if K is a nice compact superset of \(\sigma\) (S), then for any \(f\in R^{\infty}(K,\mu)\), these inclusions become equalities. An example is produced to show that \(\sigma_{ap}(f(S))\) cannot be expressed solely in terms of f and \(\sigma_{ap}(S)\). Finally, some open problems are discussed. Throughout use is made of the function theory of Chaumat, Gamelin, and Garnett. \(\{\) The following two sentences were inadvertently omitted and should begin the first paragraph on page 379: ''Consequently we may suppose that \(z_ n\in \sigma_ e(S)\) for only finitely many n. Dropping…
Mathematics Subject Classification ID: 47A60
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47B20
zbMATH DE Number: 3855803
zbMATH Keywords: spectral mapping theorems
zbMATH Keywords: subnormal operator
zbMATH Keywords: minimal normal extension
zbMATH Keywords: scalar-valued spectral measure
zbMATH Keywords: positive lower capacity density
author: James J. Dudziak
MaRDI profile type: Publication
full work available at URL: 0022-1236(84)90083-1
OpenAlex ID: W2027835484
cites work: Multiplication operators on Bergman spaces.
cites work: Weakly closed algebras of subnormal operators
cites work: Q3930259
cites work: Adhérence faible étoilé d'algèbres de fractions rationnelles
cites work: Q5618011
cites work: Distinguished Homomorphisms and Fiber Algebras
description: scientific article; zbMATH DE number 3855803