## Weighted composition operators between weighted Bergman spaces and weighted Banach spaces of holomorphic functions.(English)Zbl 1161.47018

Let $$w,v$$ be weights on the open unit disc $$\mathbb{D}$$ of the complex plane. The weighted Bergman space $$A_{w,p}$$ is the vector space of analytic functions on $$\mathbb{D}$$ such that $$\| f\| _{w,p}^p=\int_\mathbb{D} | f(z)| ^pw(z)\,dA(z)$$ is finite, where $$dA$$ denotes the normalized area measure. Let $$H^\infty_v$$ denote the vector space of analytic functions on $$\mathbb{D}$$ such that $$\| f\| _{v}=\sup_\mathbb{D} v(z) | f(z)|\,dA(z)$$ is finite. A. K. Sharma and S. D. Sharma [Commun. Korean Math. Soc. 21, No. 3, 465–474 (2006; Zbl 1160.47308)] characterized the boundedness and compactness of weighted composition operators $$\psi C_\phi:A_{w,p}\to H^\infty_v$$ for weights of the standard type $$(1-| z| ^2)^\alpha$$. Inspired by this work, the author extends such characterizations for $$w=| u|$$, with $$u$$ a non-vanishing analytic function.

### MSC:

 47B33 Linear composition operators 47B38 Linear operators on function spaces (general)

### Keywords:

weighted Bergman space; weighted composition operator

Zbl 1160.47308
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