Solvability of the functional equation $f=(T-I)h$ for vector-valued functions

Ryotaro Sato¹

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

Colloquium Mathematicum, Tome 99 (2004) no. 2, pp. 253-265

Résumé

Let $X$ be a reflexive Banach space and $({\mit \Omega },{\mathcal A},\mu )$ be a probability measure space. Let $T:M(\mu ;X)\rightarrow M(\mu ;X)$ be a linear operator, where $M(\mu ;X)$ is the space of all $X$-valued strongly measurable functions on $({\mit \Omega },{\mathcal A},\mu )$. We assume that $T$ is continuous in the sense that if $(f_{n})$ is a sequence in $M(\mu ;X)$ and $\mathop {\rm lim}_{n\rightarrow \infty } f_{n}=f$ in measure for some $f\in M(\mu ;X)$, then also $\mathop {\rm lim}_{n\rightarrow \infty } Tf_{n}=Tf$ in measure. Then we consider the functional equation $f=(T-I)h$, where $f\in M(\mu ;X)$ is given. We obtain several conditions for the existence of $h\in M(\mu ;X)$ satisfying $f=(T-I)h$.

Détail

DOI : 10.4064/cm99-2-9

Keywords: reflexive banach space mit omega mathcal probability measure space rightarrow linear operator where space x valued strongly measurable functions mit omega mathcal assume continuous sense sequence mathop lim rightarrow infty measure mathop lim rightarrow infty measure consider functional equation t i where given obtain several conditions existence satisfying t i