When is a Function of a Toeplitz Operator Close to a Toeplitz Operator?

Abstract

In this paper we consider conditions under which the operator f(Tα) - T f·α belongs to the Schatten — von Heumann class Sp and in particular conditions when f(Tα) - T f·α is of trace class. Here Tα is a Toeplitz operator which is defined for bounded α on the Hardy class H2 by $$ {T_\varphi }f = {\mathbb{P}_ + }\varphi f $$ (1) , where P+ is the orthogonal projection from L2 onto H2. We shall consider Toeplitz operators Tα for unbounded functions too. For α≈L2 the operator Tα is defined by (1) on a dense subset of H2.