Operators commuting with $L^{\mathrm{\infty }}$]{Operators commuting with $L^{\mathrm{\infty }}$ on Köthe--Bochner spaces

Let $E(X)$ be a Köthe–Bochner space of functions on a $\sigma$-finite measure space with values in a Banach space $X$. We study bounded operators on $E(X)$ that commute with the operators of multiplication by functions from $L^{\mathrm{\infty }}$. If $X$ is separable, every such operator acts pointwise by an essentially bounded $L(X)$-valued function. For a nonseparable $X$, a lifting of $L^{\mathrm{\infty }}$ provides a representation by $L(X,X^{\star \star })$-valued functions, and by $L(X)$-valued functions if $X$ is complemented in $X^{\star \star }$, with the sharp norm bound if $X$ is a dual space. As an application, we show that if the transformation is aperiodic, then the spectral projections of invertible weighted composition operators with operator-valued weights are operators of multiplication by $L(X)$-valued functions whenever $X$ is separable or complemented in $X^{\star \star }$, and that, if moreover the measure is ergodic and $\dim X=m<\mathrm{\infty }$, the spectrum of such an operator has at most $m$ connected components. Finally, for $L^p(\mu ;X)$, $1\le p<\mathrm{\infty }$, an aperiodic nonsingular transformation and a weight that is measurable in the operator norm, we prove that the spectral projections are operators of multiplication by projection-valued functions for an arbitrary Banach space $X$, for instance for $X=c_0(\mathrm{\Gamma })$ with $\mathrm{\Gamma }$ uncountable. The proof uses an inverse-closedness result for the algebra generated by norm-measurable multipliers and the composition operator.