Reproducing Kernel for The Dirichlet Space of The Upper Half - Plane and Ces_Aro Type Operator

Abstract

Reproducing kernels for spaces of analytic functions continues to be of interest to many mathematicians. Most studies have concentrated on the analytic spaces of the unit disk. Reproducing kernels for the Bergman, Hardy and Dirichlet spaces of the unit disk have been extensively determined. There has been a growing interest on the analytic spaces of the upper half plane in the recent past. For instance, the Bergman and the Szeg o kernels of the upper half plane have recently been determined. However, the theory of the Dirichlet space of the upper half plane is not well established in literature. In this study therefore, we have determined the reproducing kernel for the Dirichlet space of the upper half plane using the Cayley transform to construct an invertible isometry between the corresponding spaces of the unit disk and that of the upper half plane. By applying Cauchy-Schwarz inequality, we have established the growth condition for functions in the Dirichlet space of the upper half plane. We have then constructed an integral operator of the Ces_aro type which is acting on the Dirichlet space of the upper half plane using the approach of strongly continuous semigroups of composition operators on Banach spaces. Moreover, we have determined the spectra and norm properties of the Ces_aro type operator using the spectral mapping theorems as well as the Hille-Yosida theorem. Results of this study have contributed new knowledge to this area of mathematics and will advance further research on this and related areas