In this work, we characterize the bounded and compact weighted composition operators from a large class of Banach spaces X of analytic functions on the open unit disk into Zygmund-type spaces. Under more restrictive conditions, we provide an approximation of the essential norm of such operators. We also show that all bounded weighted composition operators from X to the little Zygmund-type space are compact and characterize such operators. We apply our results to the cases when X is the Hardy space \(H^p\) for \(1\le p\le \infty \) and the weighted Bergman space \(A_\alpha ^p\) for \(\alpha >-1\) and \(1\le p<\infty \).

中文翻译：

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从解析函数空间到Zygmund型空间的算子范数和加权合成算子的基本范数

在这项工作中，我们将有界和紧致加权合成算子的特征从开放式单位磁盘上的一大类Banach空间X的解析函数划分为Zygmund类型的空间。在更严格的条件下，我们提供了此类算子基本准则的近似值。我们还表明，从X到小Zygmund型空间的所有有界加权合成算子都是紧凑的，并且具有此类算子的特征。我们运用我们的研究结果对案件时X是强壮的空间\（H ^ P \）的\（1个\乐p \乐\ infty \）和加权Bergman空间上\（A_ \阿尔法^ P \）的\（ \ alpha> -1 \）和\（1 \ le p <\ infty \）。