Differentiation of Banach-space-valued additive processes
Volume 147 / 2001
Abstract
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 $.
