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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)

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