Spectrum of the Cesaro-Hardy operator in Lorentz Lp,q(0,1) spaces

Аңдатпа

The aim of this paper is to investigate the spectrum of the Cesaro-Hardy operator in Lorentz L_p,q spaces over (0,1). In this paper we extended Leibowitz's results for L_p space to Lorentz spaces. Note that L_p space is a special case of Lorentz spaces when indexes p and q coincide. Interestingly, we obtained the same results as for L_p space. The point spectrum is obtained by solving an Euler differential equation of first order. We used the operator P_ξ to find the resolvent set of the Cesaro-Hardy operator. This operator was defned in Boyd's work in [1]. Boundedness of the operator P_ξ on L_p was proved in the same paper. But its boundedness on L_p,q was proved in this paper by using L_p,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.