The essential norm of a composition operator on the minimal Möbius invariant space

Abstract

We derive a formula for the essential norm of a composition operator on the minimal Mobius invariant space of analytic functions. This extends a recent result due to Wulan and Xiong, and completes the picture of the situation in the Besov space setting. Our methods carry over to the case of the Bergman space A1, so we are able to complement a result of Vukoti¢ concerning the essential norm of an operator on that space. Moreover, we show that the essential norm of a non-compact composition operator is at least 1. We also obtain lower bounds depending on the behavior of the symbol near the boundary, and calculate the order of magnitude of the essential norm of composition operators induced by finite Blaschke products.