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(\mathbbT)H^p({\mathbb{T}}) over the unit circle \mathbbT{\mathbb{T}} is a connected subset of the complex plane for every bounded measurable symbol a and 1 < p < ∞. In 1972, Ronald Douglas established the connectedness of the essential spectrum of T(a) on H²(\mathbbT)H^2({\mathbb{T}}). We show that, as was suspected, these results remain valid in the setting of Hardy spaces H^p(Γ,w), 1 < p < ∞, with general Muckenhoupt weights w over arbitrary Carleson curves Γ.