Bulletin of the Australian Mathematical Society

Research Article

Composition operators on some Möbius invariant Banach spaces

Shamil Makhmutova1 and Maria Tjania2

a1 Department of Mathematics, Ufa State Aviation Technical University, Ufa 450000, Russia, e-mail: makhm@ugatu.ac.ru

a2 Department of Mathematical Sciences, 301 Science Engineering Building, University of Arkansas, Fayetteville AR 72701, United States of America, e-mail: mtjani@comp.uark.edu

Abstract

We characterise the compact composition operators from any Mobius invariant Banach space to VMOA, the space of holomorphic functions on the unit disk U that have vanishing mean oscillation. We use this to obtain a characterisation of the compact composition operators from the Bloch space to VMOA. Finally, we study some properties of hyperbolic VMOA functions. We show that a function is hyperbolic VMOA if and only if it is the symbol of a compact composition operator from the Bloch space to VMOA.

(Received July 06 1999)