Asymptotic behaviour of convolution semigroups (Q818964)

Let \(X\) be a Banach space, \(U\in L^\infty(\mathbb{R}, X)\) and let \(\mu\) be a probability measure on the Borel sets of \(\mathbb{R}\) that is not singular with respect to the Lebesgue measure. The author deduces necessary and sufficient conditions for \(U * \mu^{* n}\) to converge uniformly on the real axis. He applies the main results in order to prove an ergodic theorem for equibounded \(C_0\)-semigroups and some consequences for subordinated semigroups.