L-p contraction semigroups for vector valued functions

Let (T) over tilde(t) be a contraction semigroup on the space of vector val ued functions L-2(X, m, K) (K is a Hilbert space). In order to study the ex tension of (T) over tilde(t) to a contaction semigroup on L-p(X, m, K), 1 l ess than or equal to p < oo, Shigekawa [Sh] studied recently the domination property \(T) over tilde(t)u\(K) less than or equal to T-t\u\(K) where T-t is a symmetric sub-Markovian semigroup on L2(X, In, R). He gives in the se tting of square field operators sufficient conditions for the above inequal ity The aim of the present paper is to show that the methods of [12] and [1 3] can be applied in the present setting and provide two ways for the exten sion of (T) over tilde(t) to L-p. We give necessary and sufficient conditio ns in terms of sesquilinear forms for the L-infinity-contractivity property \\(T) over tilde(t)u\\L-infinity(X,m,K) less than or equal to \\u\\L-infin ity(X,m,K), as wen as for the above domination property in a more general s ituation.