Analytic equivalence and similarity of operators.
type: Item
instance of: scholarly article
title: Analytic equivalence and similarity of operators.
zbMATH Open document ID: 1055.47002
DOI: 10.1007/BF01193673
author: Benoît Pruvost
published in: Integral Equations and Operator Theory
publication date: 2003-07-15T00:00:00Z
review text: Two bounded linear operators \(T,R\) on a complex separable Hilbert space \(H\) are said to be analytically equivalent on an open set \(G\subset{\mathbb C}\) if there exist holomorphic maps \(A,B\) from \(G\) to the set of invertible operators such that \(T-\lambda I=A(\lambda)(R-\lambda I)B(\lambda)\) for all \(\lambda\in G\). If \(T\) and \(R\) are similar, then this can be achieved with constant \(A\) and \(B\). It then becomes natural to ask conditions under which analytic equivalence implies similarity. \textit{C. Apostol} [Oper. Theory Adv. Appl. 32, 15--35 (1988; Zbl 0655.47003)] conjectured that if \(G\supset \sigma_g(T)\cup \sigma_g(R)\), where \(\sigma_g\) denotes the generalised spectrum, then analytic equivalence of \(T\) and \(R\) on \(G\) implies similarity. In the paper under review, the author proves the conjecture in the special case where \(T\) is an isometry with \(\dim \ker(T^*)\) finite. One ingredient of the proof is the following result which is of independent interest: an operator \(R\) is similar to an isometry if and only if its eigenvalues, if any, are on the unit circle and there exist an operator \(K\) with closed range and finite-dimensional kernel and two constants \(M,m>0\) such that \(m\| Kx\| \leq\| R^nx\| \leq M\| x\| \) for all \(x\in H\) and \(n\geq 0\).
reviewed by: Line Baribeau
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47A11
Mathematics Subject Classification ID: 47A56
zbMATH DE Number: 1890564
zbMATH Keywords: operator-valued analytic functions
zbMATH Keywords: generalized spectrum
zbMATH Keywords: similarity to isometries
MaRDI profile type: Publication
cites work: Holomorphic Vector-Valued Functions on a Domain of Holomorphy
cites work: The reduced minimum modulus
cites work: Perturbation theory for nullity, deficiency and other quantities of linear operators
cites work: Analytic operator functions with compact spectrum. I. Spectral nodes, linearization and equivalence
cites work: Q4951127
cites work: Q3469757
cites work: Q5785370
cites work: Every completely polynomially bounded operator is similar to a contraction
cites work: On analytic equivalence of operator polynomials
cites work: Q4283131
cites work: Q3254471
full work available at URL: https://doi.org/10.1007/bf01193673
OpenAlex ID: W2076184667
Recommended article: Q3802086
Recommended article: Similarity of an operator to an isometric operator
Recommended article: Q3719314
Recommended article: Isometrically equivalent composition operators on spaces of analytic vector-valued functions
Recommended article: Q3468183
Recommended article: Classes of operators related to isometries
Recommended article: Q4431851
Recommended article: Power-bounded Invertible Operators and Invertible Isometries on L p Spaces
Recommended article: Q4374400
Recommended article: Q3333575
description: scientific article; zbMATH DE number 1890564