Connectedness of Spectra of Toeplitz Operators on Hardy Spaces with Muckenhoupt Weights Over Carleson Curves

Harold Widom proved in 1966 that the spectrum of a Toeplitz operator T(a) acting on the Hardy space H-p(T) over the unit circle T is a connected subset of the complex plane for every bounded measurable symbol a and 1 < p < infinity. In 1972, Ronald Douglas established the connectedness of the essential spectrum of T(a) on H-2(T). We show that, as was suspected, these results remain valid in the setting of Hardy spaces H-p(Gamma, w), 1 < p < infinity, with general Muckenhoupt weights w over arbitrary Carleson curves.