The level sets of the resolvent norm and convexity properties of Banach spaces
type: Item
instance of: scholarly article
title: The level sets of the resolvent norm and convexity properties of Banach spaces
zbMATH Open document ID: 1179.47002
author: Eugene Shargorodsky
author: S. Shkarin
published in: Archiv der Mathematik
publication date: 2009-11-23T00:00:00Z
review text: \textit{J.\,Globevnik} [Ill.\ J.\ Math.\ 20, 503--506 (1976; Zbl 0322.30040)] posed the question whether or not the norm of the resolvent of an element of a unital Banach algebra can be constant on an open subset of the resolvent set. The same question was posed as an open problem by A.\,Böttcher in 1994 at a Banach Semester in Warsaw. In the 1990s, A.\,Böttcher highlighted the importance of the question of Globevnik for computational spectral analysis. \textit{E.\,Sharogodsky} constructed in [Bull.\ Lond.\ Math.\ Soc.\ 40, No.\,3, 493--504 (2008; Zbl 1147.47007)] an invertible bounded linear operator on a Banach space and a closely densely defined operator on a Hilbert space with resolvent norms that are constant in a neighbourhood of zero, and then gives an affirmative answer to the Globevnik-Böttcher question. The main theorem of this article is that there exist a separable, reflexive and strictly convex Banach space \(X\) and an operator \(T\in L(X)\) such that \(\|(T-\lambda I)^{-1}\|\) is constant in a neighborhood of \(\lambda =0\). Some useful remarks are also made.
reviewed by: V. Lokesha
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 47A30
zbMATH DE Number: 5637242
zbMATH Keywords: norm of the resolvent
zbMATH Keywords: pseudospectrum
zbMATH Keywords: strictly convex Banach spaces
MaRDI profile type: Publication
full work available at URL: s00013-009-0001-z
OpenAlex ID: W1986685357
cites work: Pseudospectra and singular values of large convolution operators
cites work: Norms of inverses, spectra, and pseudospectra of large truncated Wiener-Hopf operators and Toeplitz matrices
cites work: Q4650329
cites work: On convexity properties of \(\psi\)-direct sums of Banach spaces.
cites work: Q3254057
cites work: On Complex Strict and Uniform Convexity
cites work: Norm-constant analytic functions and equivalent norms
cites work: Complex Convexity and Finitely Additive Vector Measures
cites work: On the level sets of the resolvent norm of a linear operator
cites work: The Strong Maximum Modulus Theorem for Analytic Functions into a Banach Space
DOI: 10.1007/S00013-009-0001-Z
description: scientific article; zbMATH DE number 5637242