Annales Academić Scientiarum Fennicć
Mathematica
Volumen 37, 2012, 203-214

THE ESSENTIAL NORM OF A COMPOSITION OPERATOR ON THE MINIMAL MÖBIUS INVARIANT SPACE

Themis Mitsis and Michael Papadimitrakis

University of Crete, Department of Mathematics
Knossos Ave., 71409 Heraklion, Greece; themis.mitsis 'at' gmail.com

University of Crete, Department of Mathematics
Knossos Ave., 71409 Heraklion, Greece; papadim 'at' math.uoc.gr

Abstract. We derive a formula for the essential norm of a composition operator on the minimal Möbius 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 Vukotic 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.

2010 Mathematics Subject Classification: Primary 47B33.

Key words: Composition operator, essential norm, Möbius invariant space.

Reference to this article: T. Mitsis and M. Papadimitrakis: The essential norm of a composition operator on the minimal Möbius invariant space. Ann. Acad. Sci. Fenn. Math. 37 (2012), 203-214.

Full document as PDF file

doi:10.5186/aasfm.2012.3714