Hyperbolic operator semigroups and Lyapunov's equation
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.
scientific article; zbMATH DE number 5980764