# zbMATH — the first resource for mathematics

Spectral conditions of mixing $$C_0$$-semigroups on Banach spaces. (English) Zbl 1299.47086
Summary: Let $$\{T(t): t\geq 0\}$$ be a $$C_0$$-semigroup on a Banach space $$X$$ and $$A$$ denote its infinitesimal generator. Using the point spectrum of $$A$$, we give sufficient conditions for $$\{T(t): t\geq 0\}$$ to be mixing or not mixing. We also characterize mixing by an admissible weight function for the translation semigroup, and employing this characterization we show that the spectral condition for $$\{T(t): t\geq 0\}$$ to be mixing is not necessary.
##### MSC:
 47D06 One-parameter semigroups and linear evolution equations 47A10 Spectrum, resolvent 47D60 $$C$$-semigroups, regularized semigroups
##### Keywords:
$$C_0$$-semigroup; mixing; point spectrum; translation semigroup