A mean value inequality with applications to Bergman space operators

Patrick Robert Ahern and Zeljko Cuckovic

Vol. 173 (1996), No. 2, 295–305

DOI: 10.2140/pjm.1996.173.295

Abstract

If u is integrable over the unit disc and u = Tu, where T is the Berezin operator then it is known that u must be harmonic. In this paper we give examples to show that the condition Tu ≥ u does not imply that u is subharmonic, but we are able to show that the condition Tu ≥ u does imply that u must be “almost” subharmonic near the boundary in an appropriate sense. We give two versions of this “almost” subharmonicity, a “pointwise” version and a “weak-star” version. We give applications of these results to hyponormal Toeplitz operators on the Bergman space.

Mathematical Subject Classification 2000

Primary: 47B38

Secondary: 46E20, 47B20, 47B35

Milestones

Received: 12 October 1993

Published: 1 April 1996