SPECTRUM OF THE GENERALIZED CESÀRO OPERATOR ON LORENTZ SPACES
Tulenov K.S. Ozbekbay B.O.
25 March 2025al-Farabi Kazakh State National University
KazNU Bulletin. Mathematics, Mechanics, Computer Science Series
2025#125Issue 166 - 80 pp.
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 $eta=1$, 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 boundMedness 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.
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Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
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