Piecewise continuous symbols

Abstract

This chapter is the heart of the book. By first employing a localization theorem and subsequently constructing a Wiener-Hopf factorization for the symbols of the local representatives, we will completely identify the essential spectra of Toeplitz operators with piecewise continuous symbols. We know from the preceding chapter that the essential spectrum of a classical Toeplitz operator is the union of the essential range of the symbol and of line segments joining the endpoints of each jump. We will show that in the general case these line segments metamorphose into circular arcs, logarithmic double spirals, horns, spiralic horns, and eventually into what we call leaves. The shape of the leaves can be described in terms of the indicator functions.