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L-p contraction semigroups for vector valued functions

Abstract : Let (T) over tilde(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 (T) over tilde(t) to a contaction semigroup on L-p(X, m, K), 1 less 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 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 (T) over tilde(t) to L-p. We give necessary and sufficient conditions 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-infinity(X,m,K), as wen as for the above domination property in a more general situation.
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Submitted on : Thursday, May 3, 2012 - 9:08:47 AM
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Em Ouhabaz. L-p contraction semigroups for vector valued functions. Positivity, Springer Verlag, 1999, 3 (1), pp.83--93. ⟨10.1023/A:1009711107390⟩. ⟨hal-00693892⟩

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