SPECTRUM OF GENERALIZED CESARO OPERATOR ON THE LORENTZ SPACES

Abstract

The aim of this paper is to investigate the boundedness and spectrum of generalized Cesaro operators defined on Lorentz spaces over a finite interval and the positive half-line. When b=1, these operators coincide with the classical Cesaro operator. In this paper, we extend the results obtained for Sobolev spaces in [5] to Lorentz spaces. The primary tools employed in this work are C₀-groups, C₀-semigroups, and their generators. C₀-groups and C₀-semigroups are used to demonstrate the boundedness of the generalized Cesaro operator. Since the spectrum of the bounded linear operators is non-empty, we investigate the spectrum of the generalized Cesaro operator. The generators of these C₀-groups and C₀-semigroups are utilized to analyze the spectral properties of the generalized Cesaro operator. We study the spectra of the generators and determine the spectra of the generalized Cesaro operators using the spectral mapping theorem. Additionally, we provide results on the point spectrum of generalized Cesaro operators defined on Lorentz spaces over a finite interval.