# Composition operators: ${N}_{\alpha }$ to the Bloch space to ${Q}_{\beta }$

Studia Mathematica (2000)

• Volume: 139, Issue: 3, page 245-260
• ISSN: 0039-3223

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## Abstract

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Let ${N}_{\alpha }$,B and Qβ be the weighted Nevanlinna space, the Bloch space and the Q space, respectively. Note that B and ${Q}_{\beta }$ are Möbius invariant, but ${N}_{\alpha }$ is not. We characterize, in function-theoretic terms, when the composition operator ${C}_{\varphi }f=f◦\varphi$ induced by an analytic self-map ϕ of the unit disk defines an operator ${C}_{\varphi }:{N}_{\alpha }\to B$, $B\to {Q}_{\beta }$, ${N}_{\alpha }\to {Q}_{\beta }$ which is bounded resp. compact.

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