Hyperbolic operator semigroups and Lyapunov's equation
type: Item
instance of: scholarly article
title: Hyperbolic operator semigroups and Lyapunov's equation
zbMATH Open document ID: 1258.47071
DOI: 10.1134/S0001434611010251
author: A. A. Vorob'ev
author: M. Yu. Romanova
author: Anatoly G. Baskakov
published in: Mathematical Notes
publication date: 2011-11-25T00:00:00Z
review text: The authors consider a complex Hilbert space \({\mathcal H}\) and the Banach algebra of linear bounded operators acting on \({\mathcal H}\), \(\operatorname{End} {\mathcal H}\). The authors extend Krein's results to the generators of strongly continuous groups of operators and to the generators of certain classes of semigroups of operators. One of the main results of the paper contains a necessary and sufficient condition for a strongly continuous group of operators \(T:{\mathbb R}\rightarrow \operatorname{End} {\mathcal H}\) to be hyperbolic, namely: there exist uniformly negative definite operators \(F\ll 0\) and \(F_* \ll 0\) from the algebra \(\operatorname{End} {\mathcal H}\) such that the equations \(A^*X+XA=F\), \(AX+XA^*=F_*\) have self-adjoint solutions \(W, W_* \in \operatorname{End} {\mathcal H}\), respectively. In this context, \(A: D(A) \subset {\mathcal H} \rightarrow {\mathcal H}\) is the generator of a hyperbolic semigroup of operators \(T: {\mathbb R}_+ \rightarrow \operatorname{End} {\mathcal H}\). Applications to differential equations with variable coefficients are given.
reviewed by: Silvia-Otilia Corduneanu
Mathematics Subject Classification ID: 47H20
zbMATH DE Number: 5980764
zbMATH Keywords: hyperbolic semigroup of operators
zbMATH Keywords: Lyapunov's equation
zbMATH Keywords: exponential dichotomy
zbMATH Keywords: Howland hyperbolic semigroup
zbMATH Keywords: Banach algebra
zbMATH Keywords: Krein's theorem
MaRDI profile type: Publication
cites work: Q4315840
cites work: Q4258125
cites work: Semigroups of difference operators in spectral analysis of linear differential operators
cites work: Finite-difference operators in the study of differential operators: solution estimates
cites work: Q4012897
cites work: How and Why to Solve the Operator Equation AX −XB = Y
cites work: Q4191141
cites work: Q5526485
cites work: Hyperbolic evolution semigroups on vector valued function spaces
full work available at URL: s0001434611010251
OpenAlex ID: W2026737978
description: scientific article; zbMATH DE number 5980764