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Generalized weighted composition operators from Bloch type spaces to weighted Bergman spaces. (English) Zbl 1130.47017
Given $$0<p<+\infty$$ and $$-1<\beta<+\infty,$$ let $$A_\beta^p$$ be the (weighted Bergman) space of all analytic functions $$f$$ in the unit disc such that $$\int_{\mathbb D}| f(z)| ^p(1-| z| ^2)^\beta \,dA(z)$$ is finite, where $$A$$ is the normalized Lebesgue measure in the unit disc. Let $${\mathcal B}^\alpha$$, $$\alpha>0,$$ be the (Bloch type) space of all analytic functions $$f$$ in the unit disc such that $$\sup_{z\in {\mathbb D}}| f'(z)| (1-| z| ^2)^\alpha$$ is finite. Both are endowed with the natural norm. For $$f\in H({\mathbb D}),$$ $$D^n_{\varphi,u}(f)=u\cdot f^{(n)}\circ \varphi$$ defines an operator by multiplication with $$u\in A_\beta^p,$$ $$n$$-times derivation and composition with an analytic self-map of the disc $$\varphi$$. The main result of the present paper asserts that if $$D^n_{\varphi,u}:{\mathcal B}^\alpha\to A_\beta^p$$ is bounded (even on the little Bloch type space $${\mathcal B}_0^\alpha$$), then it is a compact operator.

##### MSC:
 47B33 Linear composition operators 46E15 Banach spaces of continuous, differentiable or analytic functions 30H05 Spaces of bounded analytic functions of one complex variable 30D45 Normal functions of one complex variable, normal families