Exponential-cosine operator-valued functional equation in the strong operator topology
type: Item
instance of: scholarly article
title: Exponential-cosine operator-valued functional equation in the strong operator topology
zbMATH Open document ID: 0558.47032
DOI: 10.1007/BF02192668
author: Harshinder Singh
author: Harkrishan L. Vasudeva
published in: Aequationes Mathematicae
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/137021
review text: Let X be a Banach space and let B(X) denote the family of bounded linear operators on X. Let \(R^+=[0,\infty)\). A one parameter family of operators \(\{\) S(t); \(t\in R^+\}\), \(S: R^+\to B(X),\) is called exponential-cosine operator function if \(S(0)=1\) and \(S(s+t)- 2S(s)S(t)=(S(2s)-2S^ 2(s))S(t-s),\) for all \(s,t\in R^+\), \(s\leq t\). Let \(Af=\lim_{h\to 0}\frac{S(h)f-f}{h},\) \(f\in D(A)\), and \(Bf=\lim_{h\to 0}\frac{S(2h)f-2S(h)f+f}{h^ 2},\) \(f\in D(B)\). It is shown that for a strongly continuous exponential-cosine operator function \(\{\) S(t)\(\}\), \(f\in D(A^ 2)\) implies \(\int^{t}_{0}(t-u)S(u)fdu\in D(B)\) and \[ B\int^{t}_{0}(t-u)S(u)fdu=S(t)f-f+tAf- 2A\int^{t}_{0}S(u)fdu+2A^ 2\int^{t}_{0}(t-u)S(u)fdu. \] D(B) is seen to be dense in \(D(A^ 2)\). Some regularity properties of S(t) have also been obtained.
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 47D99
Mathematics Subject Classification ID: 39B99
zbMATH DE Number: 3890204
zbMATH Keywords: exponential-cosine operator function
zbMATH Keywords: strongly continuous
MaRDI profile type: Publication
cites work: Q5514010
cites work: On an exponential-cosine operator-valued functional equation
cites work: On some functional equations in Banach spaces
cites work: Q5720427
cites work: Q5535861
description: scientific article; zbMATH DE number 3890204