On closures of joint similarity orbits
type: Item
instance of: scholarly article
title: On closures of joint similarity orbits
zbMATH Open document ID: 0569.47018
DOI: 10.1007/BF01204700
author: Raúl E. Curto
author: Domingo A. Herrero
published in: Integral Equations and Operator Theory
publication date: 1985-01-01T00:00:00Z
review text: For an n-tuple \(T=(T_ 1,...,T_ n)\) of operators on a Hilbert space \({\mathcal H}\), the joint similarity orbit is \[ {\mathcal S}(T)=\{VTV^{- 1}:=(VT_ 1V^{-1},...,VT_ nV^{-1}):\quad V\quad is\quad invertible\quad on\quad {\mathcal H}\}. \] The paper is devoted to a careful study of the structure of the norm closure of \({\mathcal S}(T)\) (in \({\mathcal L}({\mathcal H})^ n)\), both in the commutative and noncommutative cases. By developing a Rota model for commuting n-tuples over the Taylor spectrum, the authors study \(\overline{{\mathcal S}(T)}\) for T quasi-nilpotent or, more generally, with totally disconnected spectrum. They then consider limits of nilpotent and normal n-tuples. By means of a number of examples, it is seen how different the several variables theory is from that for a single operator. Moreover, the noncommuting case offers many surprising facts relating the closure of \({\mathcal S}(T)\) to the Harte spectrum of T and the lack of commutativity. The paper also deals with closed joint similarity orbits; in addition, it is shown that a continuous function on \({\mathcal L}({\mathcal H})^ n\) which is invariant under joint similarity must be constant.
Mathematics Subject Classification ID: 47A55
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47B15
Mathematics Subject Classification ID: 47A65
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 47A53
zbMATH DE Number: 3909142
zbMATH Keywords: joint similarity orbit
zbMATH Keywords: Rota model for commuting n-tuples over the Taylor spectrum
zbMATH Keywords: quasi-nilpotent
zbMATH Keywords: totally disconnected spectrum
zbMATH Keywords: limits of nilpotent and normal n-tuples
zbMATH Keywords: noncommuting case
zbMATH Keywords: Harte spectrum
MaRDI profile type: Publication
cites work: Q3689716
cites work: On closures of similarity orbits of essentially nilpotent operators of order two
cites work: The closure of the similarity orbit of a Hilbert space operator
cites work: Subalgebras of C\(^*\)-algebras. II
cites work: Notes on extensions of \(C^*\)-algebras
cites work: Q3688941
cites work: Rota's Theorem for General Functional Hilbert Spaces
cites work: The Joint Spectrum of Commuting Nonnormal Operators
cites work: Models for n-tuples of noncommuting operators
cites work: The spectral mapping theorem for joint approximate point spectrum
cites work: On commuting contractions
cites work: Fredholm and Invertible n-Tuples of Operators. The Deformation Problem
cites work: Spectral Permanence for Joint Spectra
cites work: Q3749599
cites work: Generalized Bergman Kernels and the Cowen-Douglas Theory
cites work: Spectral Properties of Cyclic Subnormal m-Tuples
cites work: Joint spectra
cites work: Remark to a paper of A. E. Frazho: ''Models for noncommuting operators''
cites work: Q3764754
cites work: Models for noncommuting operators
cites work: Simultaneous similarity of matrices
cites work: An Asymptotic Double Commutant Theorem for C ∗ -Algebras
cites work: The spectral mapping theorem in several variables
cites work: Q5682842
cites work: Q4073948
cites work: Q3962026
cites work: Q3957479
cites work: A Trace Obstruction to Approximation by Block-Diagonal Nilpotents
cites work: Q5822102
cites work: Holomorphic functions of several variables. An introduction to the fundamental theory. With the assist. of Gottfried Barthel transl. by Michael Bridgland
cites work: Unitary dilations for commuting contractions
cites work: Q3688959
cites work: On the operator equation \(BX - XA = Q\)
cites work: On models for linear operators
cites work: Q3326284
cites work: A joint spectrum for several commuting operators
cites work: The analytic-functional calculus for several commuting operators
cites work: Q4147154
cites work: Q4403346
cites work: Q4102326
full work available at URL: https://doi.org/10.1007/bf01204700
OpenAlex ID: W2032549365
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