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Characterization of exponential stability of a semigroup of operators in terms of its action by convolution on vector-valued function spaces over \(\mathbb{R}_+\). (English) Zbl 0913.47033
Let \({\mathbf T}= (T(t): t\geq 0)\) be a semigroup acting on a Banach space \(X\), let \(A\) denote the infinitesimal generator and put \(\omega_0:= \inf\{\omega:\| T(t)\|\leq M\cdot e^{\omega t}\) (growth bound), for some \(M\}\) and \(S(A):= \sup(\text{Re }\lambda: \lambda\in\text{Spec }A)\) (spectral bound). If we consider the translation group on \(C_0(\mathbb{R})\), \(L^p(\mathbb{R})\) or on the torus group, then tensoring \({\mathbf T}\) with a translation group yields a new \(C_0\) semigroup, such that the growth bound and the spectral bound coincide with the original one [cf. Y. Latushkin and S. Montgomery-Smith, J. Funct. Anal. 127, No. 1, 173-197 (1995; Zbl 0878.47024)].
This striking result was generalized in various directions. The paper under review is concerned with a generalization for (vector valued) convolution semigroups: Let \({\mathbf T}\) be as above, then for \(f\in L^1_{\text{loc}}(\mathbb{R}_+, X)\) the convolution operator is defined as \(({\mathbf T}* f)(s):= \int^s_0 T(t)f(s- t)dt\). The main result (Theorem 0.1) states that \(\omega_0({\mathbf T})< \infty\) iff \({\mathbf T}* L^p(\mathbb{R}_+,X)\subseteq L^p(\mathbb{R}_+,X)\) equivalently, iff \({\mathbf T}* C_0(\mathbb{R}_+, X)\subseteq C_0(\mathbb{R}, X)\).
In Section 2 some applications are considered, in Section 3 an analogous characterization of \(\omega_0(T)< 0\) is obtained for (rearrangement invariant) Banach function spaces and in Section 4 similar results for the weak topology are proved.
Reviewer: W.Hazod (Dortmund)

47D06 One-parameter semigroups and linear evolution equations
47A10 Spectrum, resolvent
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