Spectrum of the Cesaro-Hardy operator in Lorentz Lp,q(0,1) spaces
DOI:
https://doi.org/10.26577/JMMCS.2023.v117.i1.04Keywords:
Cesa ́ro-Hardy operator, spectrum, point spectrum, Lorentz Lp,q spaces, rearrangement invariant spacesAbstract
The aim of this paper is to investigate the spectrum of the Cesaro-Hardy operator in Lorentz Lp,q spaces over (0,1). In this paper we extended Leibowitz's results for Lp space to Lorentz spaces. Note that Lp space is a special case of Lorentz spaces when indexes p and q coincide. Interestingly, we obtained the same results as for Lp space. The point spectrum is obtained by solving an Euler differential equation of first order. We used the operator Pξ to find the resolvent set of the Cesaro-Hardy operator. This operator was defned in Boyd's work in [1]. Boundedness of the operator Pξ on Lp was proved in the same paper. But its boundedness on Lp,q was proved in this paper by using Lp,q norm of the dilation operator. Here, we also used the Boyd's theorem, which describes boundedness of operators on rearrangement invariant spaces. We verified conditions of Boyd's theorem. It allows us to obtain a bounded inverse of the operator λI −C for some complex numbers λ.
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