Composition operators between Nevanlinna classes and Bergman spaces with weights. (English) Zbl 0996.47031

Jarchow, Hans; Xiao, Jie

J. Oper. Theory 46, No. 3, 605-618 (2001).

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)

Keywords:

order boundedness; weighted Nevanlinna classes; weighted Bergman spaces; composition operators; spaces of analytic functions