Spectrum of invertible weighted composition operators on the unit disk

Abstract

The spectrum of invertible weighted composition operators $u{C}_{\phi }$ acting on classical Banach spaces of holomorphic functions in the unit disk $D$ has been studied intensively over the years. Complete descriptions of that spectrum have been given in the elliptic or parabolic cases, that is, for φ either elliptic or parabolic, but only partial results have been obtained in the remaining case, that is, for hyperbolic φ. In this paper, we give the spectrum and the essential spectrum of $u{C}_{\phi }$ for hyperbolic φ. Our results answer in the positive several conjectures posed by different authors.

In order to deal with the above questions, we present new techniques which involve the embedding of the weight u into a cocycle ${\left(u_t\right)}_{t\in R}$ associated to an hyperbolic flow ${\left({\phi }_t\right)}_{t\in R}$. We also provide information about the range spaces and null spaces of $\lambda -u{C}_{\phi }$ for λ lying in the interior of $\sigma \left(u{C}_{\phi }\right)$.