Lp Contraction Semigroups for Vector Valued Functions

Lp Contraction Semigroups for Vector Valued Functions 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Lp Contraction Semigroups for Vector Valued Functions

Positivity , Volume 3 (1) – Oct 22, 2004
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Publisher
Kluwer Academic Publishers
Copyright
Copyright © 1999 by Kluwer Academic Publishers
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1023/A:1009711107390
Publisher site
See Article on Publisher Site

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.

Journal

PositivitySpringer Journals

Published: Oct 22, 2004

References

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