Composition operators between Nevanlinna classes and Bergman spaces with weights

Jarchow, H; Xiao, J (2001). Composition operators between Nevanlinna classes and Bergman spaces with weights. Journal of Operator Theory, 46(3, suppl.):605-618.

Abstract

We investigate composition operators between spaces of analytic functions on the unit disk $\De$ in the complex plane. The spaces we consider are the weighted Nevanlinna class $\cN_\al$, which consists of all analytic functions $f$ on $\De$ such that $\int\limits_\De\log^+ |f(z)|(1-|z|^2)^\al {\, {\rm d}x \, {\rm d}y}<\iy$, and the corresponding weighted Bergman spaces $\cA^p_\al$, $-1<\al<\iy$, $0-1$, $0<q<\iy$. We characterize, in function theoretic terms, when the composition operator $\Cf:f\mt f\ci\vf$ induced by an analytic function $\vf:\De\to\De$ defines an operator $X\to Y$ which is continuous, respectively compact, respectively order bounded.

Abstract

We investigate composition operators between spaces of analytic functions on the unit disk $\De$ in the complex plane. The spaces we consider are the weighted Nevanlinna class $\cN_\al$, which consists of all analytic functions $f$ on $\De$ such that $\int\limits_\De\log^+ |f(z)|(1-|z|^2)^\al {\, {\rm d}x \, {\rm d}y}<\iy$, and the corresponding weighted Bergman spaces $\cA^p_\al$, $-1<\al<\iy$, $0-1$, $0<q<\iy$. We characterize, in function theoretic terms, when the composition operator $\Cf:f\mt f\ci\vf$ induced by an analytic function $\vf:\De\to\De$ defines an operator $X\to Y$ which is continuous, respectively compact, respectively order bounded.