Differentiation of Banach-space-valued additive processes

Volume 147 / 2001

Ryotaro Sato Studia Mathematica 147 (2001), 131-153 MSC: 47A35, 47D03, 46E30, 46E40. DOI: 10.4064/sm147-2-3

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 $.

Authors

  • Ryotaro SatoDepartment of Mathematics
    Faculty of Science
    Okayama University
    Okayama, 700-8530 Japan
    e-mail

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