On the spectrum of cosine operator functions
type: Item
instance of: scholarly article
title: On the spectrum of cosine operator functions
zbMATH Open document ID: 0696.47044
DOI: 10.1007/BF01194559
author: Carlos Lizama
published in: Integral Equations and Operator Theory
publication date: 1989-01-01T00:00:00Z
review text: Let C(t), \(t\in {\mathbb{R}}\), be a strongly continuous cosine function, defined in a Hilbert space H, with infinitesimal generator A. Let \(\rho\) (A) and \(\sigma\) (A) denote the resolvent set and the spectrum of A respectively. It is known that for certain classes of cosine functions, including the class of uniformly bounded cosine functions, the following equivalence is true: \[ \mu \in \rho (C(t))\quad \Leftrightarrow \quad \{\lambda^ 2| ch \lambda t=\mu \}\subseteq \rho (A)\quad and\quad \sup_{ch \lambda t=\mu}\| \lambda (\lambda^ 2-A)^{-1}\| <\infty. \] In this paper the author proves that this is actually true for every strongly continuous cosine function defined in a Hilbert space.
Mathematics Subject Classification ID: 47D99
Mathematics Subject Classification ID: 47D03
zbMATH DE Number: 4140604
zbMATH Keywords: strongly continuous cosine function
zbMATH Keywords: infinitesimal generator
zbMATH Keywords: resolvent
zbMATH Keywords: spectrum
zbMATH Keywords: uniformly bounded cosine functions
reviewed by: Ali-Akbar Jafarian
MaRDI profile type: Publication
cites work: Spectral properties of cosine operator functions
cites work: Q3678037
cites work: Spectral Theory for Contraction Semigroups on Hilbert Space
cites work: Q3319291
cites work: Q3943501
cites work: A Cosine Functional Equation in Hilbert Space
cites work: Q5182164
cites work: On the Spectrum of C 0 -Semigroups
cites work: Q5535861
cites work: Cosine families and abstract nonlinear second order differential equations
description: scientific article; zbMATH DE number 4140604