L^p Contraction Semigroups for Vector Valued Functions

Abstract

Let $\tilde T_t$ be a contraction semigroup on the space of vector valued functions $L^2 (X,m,K)$ ($K$ is a Hilbert space). In order to study the extension of $\tilde T_t$ to a contaction semigroup on $L^p (X,m,K)$, $1 \leqslant p{\text{ < }}\infty$ Shigekawa [Sh] studied recently the domination property $|\tilde T_t u|_K \leqslant T_t |u|_K$ where $T_t$ is a symmetric sub-Markovian semigroup on $L^2 (X,m,\mathbb{R})$. He gives in the setting of square field operators sufficient conditions for the above inequality. The aim of the present paper is to show that the methods of [12] and [13] can be applied in the present setting and provide two ways for the extension of ${\tilde T}$ to $L^p$ We give necessary and sufficient conditions in terms of sesquilinear forms for the $L^\infty -$contractivity property $||\tilde T_t u||_{L^\infty } (X,m,K) \leqslant ||u||_{L^\infty } (X,m,K),$ as well as for the above domination property in a more general situation.