Stability of semigroups of operators and spectral subspaces
A strongly continuous semigroup \(\{e^{tA}\}_{t\geq 0}\) on a Banach space \(X\) is called stable if \[ \lim_{t\to\infty}e^{tA}x=0\;\;\forall x\in X. \] For \(x\in X\), the local resolvent set \(\rho(x,A)\) is the set of all \(\lambda\in{\mathbb C}\) such that there exists an open neighborhood \(U\) of \(\lambda\) and a \(D(A)\)-valued analytic function \(r\) such that \((\lambda - A)r(\lambda)=x\) for all \(\lambda\in U\). The complement of \(\rho(x,A)\) is a local spectrum \(\sigma(x,A)\). The local spectral subspace \(X(A,\Omega)\) is the set \(\{x\mid \sigma(x,A)\subset\Omega \}\). In the paper under review, the authors find a local spectral characterization of stability. The following sufficient conditions are proved: (1) \(X^\star(A^\star,i{\mathbb R})\) is trivial. (2) \(\sigma_{\text{point}}(A^\star)\cap i{\mathbb R}\) is empty and for all \(x^\star\in X^\star(x^\star,i{\mathbb R})\), \(\sigma(x^\star,A^\star)\) is countable. If \(A^\star\) has no eigenvalues in \((-\infty,0)\), then stability is equivalent to the following condition: For all nontrivial \(x^\star\) in \(\bigcap_{c>0,n\in{\mathbb N}} \text{Ran}((s+A^\star)^n)\), \(\{s^n\| (s+A^\star)^{-n}\| \mid s>0,n\in{\mathbb N} \}\) is unbounded.
local spectral characterization of stability
scientific article; zbMATH DE number 1761132