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Weighted composition operators between different weighted Bergman spaces and different Hardy spaces. (English) Zbl 1147.47021

The boundedness and compactness of composition operators \(C_\varphi f = f \circ \varphi\), restricted to Hardy spaces \(H^p\) and weighted Bergman spaces \(L_a^{p, \alpha}\) of the disk, have been extensively studied by many authors. M. D. Contreras and A. G. Hernandez-Diaz [Integral Equations Oper. Theory 46, No. 2, 165–188 (2003; Zbl 1042.47017)] considered weighted composition operators \((uC_\varphi)f=u(z)C_\varphi f(z)\). In the present paper, the authors use the generalized Berezin transform and related integral operators to characterize bounded and compact weighted composition operators mapping \(L_a^{p, \alpha}\) into \(L_a^{q, \beta}\) and \(H^q\) and \(H^p\). As one would expect, the results are different for the \(p \leq q\) case and the \(q<p\) case. The essential norms of these operators are also estimated. As applications, they characterize bounded and compact pointwise multiplication operators between Bergman spaces and estimate their essential norms.

MSC:

47B33 Linear composition operators
46E15 Banach spaces of continuous, differentiable or analytic functions
30D55 \(H^p\)-classes (MSC2000)

Citations:

Zbl 1042.47017
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