On the spectrum of cosine functions
type: Item
instance of: scholarly article
title: On the spectrum of cosine functions
zbMATH Open document ID: 0921.34073
author: E. Schüler
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-06-13T00:00:00Z
review text: The author gives some characterizations of the resolvent set \(\rho(A)\) of the generator \(A\) of a strongly continuous cosine function \(C(t)\) with the aid of the equation \[ u''(t)=Au(t)+f(t). \] One of these theorems shows that \(1\in\rho(C(1))\) if and only if, for every \(1\)-periodic function \(f\in C([0,1],X)\) (\(X\) is a Banach space), the above equation has a unique \(1\)-periodic mild solution of class \(C^1\). In case of a Hilbert space \(X\), the above equation has a unique \(1\)-periodic mild solution for any \(1\)-periodic function \(f\in C([0,1],X)\), if and only if, the associated sine function \(S(t)\) has the property that \(S(1)\) is invertible.
Mathematics Subject Classification ID: 34L15
Mathematics Subject Classification ID: 47A25
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 1251421
zbMATH Keywords: spectrum
zbMATH Keywords: cosine functions
zbMATH Keywords: resolvent set
zbMATH Keywords: generator
zbMATH Keywords: strongly continuous cosine function
zbMATH Keywords: associated sine function
reviewed by: Narcisa Apreutesei
MaRDI profile type: Publication
full work available at URL: jmaa.1998.6140
OpenAlex ID: W2056648560
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DOI: 10.1006/JMAA.1998.6140
description: scientific article; zbMATH DE number 1251421