Differentiation of Banach-space-valued additive processes

Ryotaro Sato ¹

¹ Department of Mathematics Faculty of Science Okayama University Okayama, 700-8530 Japan

Studia Mathematica, Tome 147 (2001) no. 2, pp. 131-153

Résumé

Let $X$ be a Banach space and $({\mit \Omega } ,{\mit \Sigma } ,\mu )$ be a $\sigma $-finite measure space. Let $L$ be a Banach space of $X$-valued strongly measurable functions on $({\mit \Omega } ,{\mit \Sigma } ,\mu )$. We consider a strongly continuous $d$-dimensional semigroup $T=\{ T(u):u=(u_{1},\mathinner {\ldotp \ldotp \ldotp },u_{d}),\ u_{i}>0$, $1\leq i\leq d\} $ of linear contractions on $L$. We assume that each $T(u)$ has, in a sense, a contraction majorant and that the strong limit $T(0)=\hbox {strong-lim}_{u\rightarrow 0}T(u)$ exists. Then we prove, under some suitable norm conditions on the Banach space $L$, that a differentiation theorem holds for $d$-dimensional bounded processes in $L$ which are additive with respect to the semigroup $T$. This generalizes a differentiation theorem obtained previously by the author under the assumption that $L$ is an $X$-valued $L_{p}$-space, with $1\leq p\infty $.

Détail

DOI : 10.4064/sm147-2-3

Keywords: banach space mit omega mit sigma sigma finite measure space banach space x valued strongly measurable functions mit omega mit sigma consider strongly continuous d dimensional semigroup mathinner ldotp ldotp ldotp leq leq linear contractions assume each has sense contraction majorant strong limit hbox strong lim rightarrow exists prove under suitable norm conditions banach space differentiation theorem holds d dimensional bounded processes which additive respect semigroup generalizes differentiation theorem obtained previously author under assumption x valued space leq infty