Local spectral properties of generators of \(C_0\)-groups and Bernstein type inequalities
type: Item
instance of: scholarly article
title: Local spectral properties of generators of \(C_0\)-groups and Bernstein type inequalities
zbMATH Open document ID: 1185.47042
author: C. Temel
author: H. S. Mustafayev
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-06-23T00:00:00Z
review text: In this paper, the authors study the local spectral properties of generators of \(C_0\) groups. Let \(T=\{T(t)\}_{t\in\mathbb R}\) be a \(C_0\) group on a complex Banach space \(X\) dominated by a weight function \(\omega(t)=(1+|t|)^{\alpha}\) (\(0\leq \alpha \leq 1\)) and let \(A\) be its generator with domain \(D(A)\). Among other things, it is proved that, if the operator \(A\) has compact local spectrum at a point \(x\in X\), then \(x\in D(A)\) and there exist double sequences of real numbers \((c_n)_{n\in\mathbb Z}\) and \((t_n)_{n\in\mathbb Z}\) such that \[ Ax=\sum_{n\in\mathbb Z}c_nT(t_n)x, \] where \( \sum_{n\in\mathbb Z}|c_n|=r_A(x)\), the local spectral radius of \(A\) at \(x\). As applications, the authors give several inequalities of Bernstein type in \(L^p\)-spaces.
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 47A11
Mathematics Subject Classification ID: 47D06
zbMATH DE Number: 5568991
zbMATH Keywords: \(C_0\)-group
zbMATH Keywords: local spectrum
zbMATH Keywords: Beurling spectrum
zbMATH Keywords: \(L^p\)-space
zbMATH Keywords: Bernstein inequality
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.04.017
OpenAlex ID: W1972719484
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DOI: 10.1016/J.JMAA.2009.04.017
description: scientific article; zbMATH DE number 5568991