Toeplitz operators with PC symbols on general Carleson Jordan curves with arbitrary Muckenhoupt weights

Abstract

We describe the spectra and essential spectra of Toeplitz operators with piecewise continuous symbols on the Hardy space H p ( Γ , ω ) H^p(\Gamma ,\omega ) in case 1 > p > ∞ 1>p>\infty , Γ \Gamma is a Carleson Jordan curve and ω \omega is a Muckenhoupt weight in A p ( Γ ) A_p(\Gamma ) . Classical results tell us that the essential spectrum of the operator is obtained from the essential range of the symbol by filling in line segments or circular arcs between the endpoints of the jumps if both the curve Γ \Gamma and the weight are sufficiently nice. Only recently it was discovered by Spitkovsky that these line segments or circular arcs metamorphose into horns if the curve Γ \Gamma is nice and ω \omega is an arbitrary Muckenhoupt weight, while the authors observed that certain special so-called logarithmic leaves emerge in the case of arbitrary Carleson curves with nice weights. In this paper we show that for general Carleson curves and general Muckenhoupt weights the sets in question are logarithmic leaves with a halo, and we present final results concerning the shape of the halo.