Compact Composition Operators on the Bloch Space and the Growth Space of the Upper Half-Plane

Abstract

In this paper, we prove that unlike Hardy and weighted Bergman spaces of the upper half-plane, there are non-trivial analytic self-maps of the upper half-plane that induce compact composition operators on the Bloch space of the upper half-plane. Moreover, we also prove that like Hardy and weighted Bergman spaces of the upper half-plane, the growth space of the upper half-plane does not support compact composition operators.