标题： 弱零最大值和乘法算子族的积分 显示英文标题

标题： Weak null maximum and integration of families of multiplication operators

摘要： 设$X$为复平面上单位圆盘上的自反 Hardy 空间或加权 Bergman 空间。 对于$X$上的有界线性算子$S$，令 $\textrm{wem}(S):= \sup_{(f_n)} \limsup_n \|Sf_n\|$，即$n\mapsto \|S f_n\|$的聚点的上确界，其中$(f_n)$为任意单位范数的弱零序列。 该量在自反加权 Bergman 空间上与本质范数一致。 对于单位圆盘上的一类合适的有界解析函数$\{ g_t : t\in]0,1[ \}$，我们描述了何时可以交换乘法算子$M_{g_t}$在$t$上的积分和$\textrm{wem}(\cdot)$，即当$\textrm{wem}( \int M_{g_t}\, dt ) = \int \textrm{wem}( M_{g_t} ) \, dt $；当函数$g_t,t\in]0,1[$可以连续地延拓到单位圆时，我们得到一个简洁的函数理论特征。 显示英文摘要

摘要： 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 $\textrm{wem}(S):= \sup_{(f_n)} \limsup_n \|Sf_n\|$, that is, the supremum of cluster points of $n\mapsto \|S f_n\|$, 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 $\textrm{wem}(\cdot)$ and integration over $t$ of the multiplication operators $M_{g_t}$, that is, when $\textrm{wem}( \int M_{g_t}\, dt ) = \int \textrm{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.