Toeplitz operators in several complex variables

Abstract

Let S be the unit sphere in C n . We investigate the properties of Toeplitz operators on S, i.e., operators of the form Tφf = P( φf) where φ ϵ L ∞( S) and P denotes the projection of L 2( S) onto H 2( S). The aim of this paper is to determine how far the extensive one-variable theory remains valid in higher dimensions. We establish the spectral inclusion theorem, that the spectrum of T φ contains the essential range of φ, and obtain a characterization of the Toeplitz operators among operators on H 2( S) by an operator equation. Particular attention is paid to the case where φ ϵ H ∞( S) + C( S) where C( S) denotes the algebra of continuous functions on S. Finally we describe a class of Toeplitz operators useful for providing counterexamples—in particular, Widom's theorem on the connectedness of the spectrum fails when n > 1.