Stabilities of Hilbert space contraction semigroups revisited (Q734978)

The authors study conditions for the exponential stability of contraction semigroups in Hilbert spaces. The main result obtained in the paper is the following: A contraction semigroup \((T(t))_{t\geq 0}\) with generator \(A\) in a Hilbert space is exponentially stable if and only if \(\alpha \| x\| ^2 \leq -\text{Re}\langle Ax;x\rangle\) for every \(x\) in the domain \(D(A)\) of the generator \(A\), and for some \(\alpha >0\). The proof of this theorem is based on a characterization of exponential stability of semigroups in Hilbert spaces due to \textit{R.\,Datko} [J.~Math.\ Anal.\ Appl.\ 32, 610--616 (1970; Zbl 0211.16802)].