Spectral sets of generalized Hausdorff matrices on spaces of holomorphic functions on [formula omitted

Abstract

Here, we study a family of bounded operators H, acting on Banach spaces of holomorphic functions X↪O(D), which are subordinated in terms of a C0-semigroup of weighted composition operators (vtCϕt), i.e., H=∫0∞vtCϕtdν(t) in the strong sense for some Borel measure ν. This family of operators extends the so-called generalized Hausdorff operators. We obtain the spectrum, point spectrum and essential spectrum of H under mild assumptions on (vtCϕt),ν and X. In particular, we obtain these spectral sets for a wide family of generalized Hausdorff operators acting on Hardy spaces, weighted Bergman spaces, weighted Dirichlet spaces and little Korenblum classes. The description for the spectra of the infinitesimal generator of (vtCϕt) is the key for our findings.