A Gleason-Kahane-\.Zelazko Theorem for $H^p_{\alpha, \beta}$ spaces

Abstract

We study two entities that have proved to be of interest and importance in their own right viz. the $H^p_{\alpha, \beta}$ spaces and the classical Gleason-Kahane-\.Zelazko (GKZ) theorem. We establish a GKZ-type theorem on the $H^p_{\alpha, \beta}$ spaces by identifying a natural class of functions serving as the counterpart of invertible elements, and proving that every continuous linear functional nonvanishing on this class is a point evaluation. As an application, we characterize weighted composition operators on these spaces. It must be noted that the $H^p_{\alpha, \beta}$ spaces do not possess the various structural advantages of the classical Hardy spaces where the GKZ theorem already exists and thus we have had to modify and establish methods that rely on suitable extensions and refinements of the known techniques. Additionally, we provide examples that illustrate the natural class of functions arising in our GKZ-type theorem and demonstrate the sharpness of our characterization of weighted composition operators under the assumption of surjectivity.