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Positivity

, Volume 3, Issue 1, pp 83–93 | Cite as

Lp Contraction Semigroups for Vector Valued Functions

  • El Maati Ouhabaz
Article

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.

Positive and \(L^\infty\)-contractive semigroups domination sesquilinear forms 

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Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • El Maati Ouhabaz
    • 1
  1. 1.Equipe d'Analyse et de Mathématiques AppliquéesUniversité de Marne-La-ValléeMarne-la- Vallée Cedex 2France

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