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Abstract

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 $f=(T-I)h$.