The Essential Spectrum of Some Toeplitz Operators

Abstract

The localization techniques of Douglas and Sarason are used to obtain the essential spectrum of the Toeplitz operator <!-- MATH ${T_\varphi }$ --> for which is the product of a continuous function and the characteristic function of a measurable subset of the unit circle. Examples are given of Toeplitz operators with one-dimensional self-commutator whose essential spectrum is the unit disk. Using an example of J. E. Brennan, the authors show the existence of a completely nonnormal, subnormal operator whose adjoint has no point spectrum.