Exponential stability of operators and operator semigroups
Extending earlier results of Datko, Pazy, and Littman on \(C_0\)- semigroups, and of Przyluski and Weiss on operators, the author proves the following: Let \(T\) be a bounded linear operator on a Banach space \(X\) and let \(r(T)\) denote its spectral radius. Let \(E\) be a Banach function space over \(\mathbb{N}\) with the property that \(\lim_{n\to \infty} |\chi_{\{0, \dots, n-1\}} |_E= \infty\). If for each \(x\in X\) and \(x^*\in X^*\) the map \(n\mapsto \langle x^*, T^n x\rangle\) belongs to \(E\), then \(r(T) <1\). By applying this to Orlicz spaces \(E\), he obtains the following result. Let \(T\) be a bounded linear operator on a Banach space \(X\) and let \(\varphi: \mathbb{R}_+\to \mathbb{R}_+\) be a non-decreasing function with \(\varphi (t)>0\) for all \(t>0\). If \(\sum_{n=0}^\infty \varphi( |\langle x^*, T^n x\rangle |)< \infty\) for all \(|x|, |x^* |\leq 1\), then \(r(T)<1\). Assuming a \(\Delta_2\)-condition on \(\varphi\), a further improvement is obtained. For locally bounded semigroups \(\mathbb{T}= \{T(t) \}_{t\geq 0}\), he obtains similar results in terms of the maps \(t\mapsto |T(t) x|\).
scientific article; zbMATH DE number 772872