SPECTRUM OF GENERALIZED CESARO OPERATOR ON THE LORENTZ SPACES

The aim of this paper is to investigate the boundedness and spectrum of generalized Ces\`{a}ro operators defined on Lorentz spaces over a finite interval and the positive half-line. When $\beta=1$, these operators coincide with the classical Ces\`{a}ro operator. In this paper, we extend the results obtained for Sobolev spaces in \cite{Lizama} to Lorentz spaces. The primary tools employed in this work are $C_0$-groups, $C_0$-semigroups, and their generators. $C_0$-groups and $C_0$-semigroups are used to demonstrate the boundedness of the generalized Ces\`{a}ro operator. Since the spectrum of the bounded linear operators is non-empty, we investigate the spectrum of the generalized Ces\`{a}ro operator. The generators of these $C_0$-groups and $C_0$-semigroups are utilized to analyze the spectral properties of the generalized Ces\`{a}ro operator.We study the spectra of the generators and determine the spectra of the generalized Ces\`{a}ro operators using the spectral mapping theorem. Additionally, we provide results on the point spectrum of generalized Ces\`{a}ro operators defined on Lorentz spaces over a finite interval.