A class of integral operators on the Dirichlet space of the upper half plane

Abstract

Exploring integral operators and spaces of analytic functions remains a focal point for many mathematicians. While majority of the research has centered around the analytic spaces of the unit disk. There has been a growing interest in extending these studies to analytic spaces of the upper half plane. The properties of integral operators on Hardy and Bergman spaces of the upper half plane have largely been determined. However, there is still more to explore regarding the Dirichlet space of the upper half-plane. In this study, we have focused on the properties of integral operators on Dirichlet spaces of the upper half plane D(U) . Specifically, we have determined the semigroup properties of composition semigroup; constructed a class on integral operators and investigated its properties on the Dirichlet space of the upper half plane. Applying methods similar to those utilized in the related works of Agwang’ and Oyugi, we identified the semigroup properties. We have constructed a class of integral operators which is acting on the Dirichlet space of the upper half plane employing the approach of strongly continuous semigroup of composition operators on Banach spaces. By employing the spectral mapping theorems and Hille-Yosida theorem we successfully identified the spectra and the norm properties of integral operators. The findings of this research contributes meaningfully to the existing literature, providing valuable insights for pure mathematicians advancing in this area. v