\(C^*\)-algebras generated by subnormal operators
type: Item
instance of: scholarly article
title: \(C^*\)-algebras generated by subnormal operators
zbMATH Open document ID: 0665.47022
DOI: 10.1016/0022-1236(88)90020-1
author: Paul M. McGuire
published in: Journal of Functional Analysis
publication date: 1988-01-01T00:00:00Z
review text: Let H be a separable complex Hilbert space, and, as usually, let B(H) be the algebra of all bounded linear operators on H. The author considers the \(C^*\)-algebra \(C^*(S)\) generated by an irreducible, essentially normal, subnormal operator \(S\in B(H)\), and he denotes a minimal normal extension of S by N. In the first section the author characterizes the \(C^*\)-algebra \(C^*(S)\) and determines the generators of \(C^*(S)\). Using the theory of \textit{L. G. Brown, R. G. Douglas} and \textit{P. A. Fillmore} [Lect. Notes Math. 345, 58-128 (1973; Zbl 0277.46053)] the author states a necessary and sufficient condition on an operator \(A\in B(H)\) in order that the \(C^*\)-algebra \(C^*(A)\) generated by A is spatially \({}^*\)-isomorphic to \(C^*(S)\), i.e. \(C^*(A)\cong C^*(S)\), this is the case if and only if the following conditions on \(A\in B(H)\) holds i) A is an irreducible, essentially normal operator on H; ii) for every continuous function f on the spectrum \(\sigma\) (N), i.e. \(f\in C^ 0(\sigma (N))\), such that the restriction of f to the essential spectrum \(\sigma_ e(S)\) of S is one-to-one, one has (a) \(\sigma_ e(A)=f(\sigma_ e(S))\), and (b) index (A-\(\lambda)=index (f(S)-\lambda)\) for every \(\lambda \in C\setminus \sigma_ e(A).\) The authors' next step is the computation of the essential spectrum \(\sigma_ e(S)\) and the index of f(S) for every \(f\in C^ 0(\sigma (N))\) with \(f|_{\sigma_ e(S)}\) is one-to-one. Some enlighting examples of irreducible, subnormal operators S with a prescribed spectral picture and the \(C^*\)-algebras \(C^*(S)\) generated by them are given. These examples leads the author to a classification of the \(C^*\)- algebras \(C^*(A)\) generated by irreducible, essentially normal operators A with essential spectrum \(\sigma_ e(A)\) equal to a finite union of disjoint Jordan curves and index (A-\(\lambda)\)\(\neq 0\) \((\lambda \in C\setminus \sigma_ e(A)\) suitable). Finally the author considers two operators \(A,B\in B(H…
Mathematics Subject Classification ID: 47B20
Mathematics Subject Classification ID: 46L05
zbMATH DE Number: 4088336
zbMATH Keywords: \(C^*\)-algebra generated by an irreducible, essentially normal, subnormal operator
zbMATH Keywords: algebra of all bounded linear operators
zbMATH Keywords: minimal normal extension
zbMATH Keywords: essentially normal operator
zbMATH Keywords: prescribed spectral picture
zbMATH Keywords: hyponormal operators with rank 1 self-commutators
MaRDI profile type: Publication
full work available at URL: 0022-1236(88)90020-1
OpenAlex ID: W2018153814
cites work: Toeplitz Operators in Multiply Connected Regions
cites work: Q4112980
cites work: Multiplication operators on Bergman spaces.
cites work: Toeplitz Operators on Bergman Spaces
cites work: Q4403336
cites work: An Invariant for Certain Operator Algebras
cites work: Seminormal Operators with Compact Self-Commutators
cites work: The C ∗ -Algebra Generated by an Isometry. II
cites work: Q3684689
cites work: Operators with C ∗ -Algebra Generated by a Unilateral Shift
cites work: Spectral properties of certain operators on Hardy spaces of planar regions
cites work: Q5618011
cites work: Extensions of the Berger-Shaw Theorem
cites work: Q4195448
cites work: Subnormal Operators, Toeplitz Operators and Spectral Inclusion
cites work: A local functional calculus
cites work: Algebras Generated by a Subnormal Operator
cites work: The analytic model of a hyponormal operator with rank one self-commutator
cites work: Q3324501
cites work: Q3903565
cites work: Analytic Functions of Class H p
cites work: Q4405392
cites work: Q3259806
cites work: Subnormal operator with a cyclic vector
description: scientific article; zbMATH DE number 4088336