Weak null maximum and integration of families of multiplication operators

Abstract

Let $X$ be a reflexive Hardy space or weighted Bergman space on the unit disk in the complex plane. For a bounded linear operator $S$ on $X$, let $\text{wem}(S):={\sup }_{(f_n)}{\mathrm{limsup}}_n\Vert S{f}_n\Vert$, that is, the supremum of cluster points of $n\mapsto \Vert S{f}_n\Vert$, where $(f_n)$ is any unit norm weakly null sequence. This quantity coincides with the essential norm on the reflexive weighted Bergman spaces. For a suitable family $\{g_t:t\in ]0,1[\}$ of bounded analytic functions on the unit disk, we characterize when one can exchange $\text{wem}(\cdot )$ and integration over $t$ of the multiplication operators $M_{g_t}$, that is, when $\text{wem}(\int M_{g_t}dt)=\int \text{wem}(M_{g_t})dt$; when the functions $g_t,t\in ]0,1[$ can be continuously extended to the unit circle, we obtain a neat function-theoretic characterization.

7 pages, several typos fixed in v2