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Composition operators between Nevanlinna classes and Bergman spaces with weights. (English) Zbl 0996.47031
Summary: We investigate composition operators between spaces of analytic functions on the unit disk $$\Delta$$ in the complex plane. The spaces we consider are the weighted Nevanlinna class $${\mathcal N}_\alpha$$, which consists of all analytic functions $$f$$ on $$\Delta$$ such that $$\int_\Delta\log^+|f(z)|(1-|z|^2)^\alpha dx dy<\infty$$, and the corresponding weighted Bergmann spaces $${\mathcal A}_\alpha^p$$, $$1-<\alpha<\infty$$, $$0<p<\infty$$. Let $$X$$ be any of the spaces $${\mathcal A}_\alpha^p$$, $${\mathcal N}_\alpha$$ and $$Y$$ any of the spaces $${\mathcal A}_\beta^q$$, $${\mathcal N}_\beta$$, $$\beta>-1$$, $$0<q<\infty$$. We characterize, in function theoretic terms, when the composition operator $$C_\varphi:f\mapsto f\circ\varphi$$ induced by an analytic function $$\varphi:\Delta\to\delta$$ defines an operator $$X\to Y$$ which is continuous, respectively compact, respectively order bounded.

##### MSC:
 47B33 Linear composition operators 46E15 Banach spaces of continuous, differentiable or analytic functions 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory 47B38 Linear operators on function spaces (general)