Spectral Properties of the Compact Rhaly and Compact Generalised Ces{à}ro Operators on Weighted $c_0$ Spaces
Jyoti Rani, Arnab Patra
TL;DR
The paper addresses the spectral analysis of Rhaly operators $R_a$ and discrete generalized Cesàro operators $C_t$ on weighted null sequence spaces $c_0(s)$. It develops boundedness and compactness criteria and derives complete fine-spectrum decompositions, including point, continuous, and residual spectra, alongside Goldberg-type classifications. For compact $R_a$, the spectrum is $\sigma(R_a,c_0(s)) = S\cup\{0\}$ with $\sigma_p= S$ and $\sigma_c=\{0\}$, while $C_t$ (for $0<t<1$) has $\sigma_p(C_t,c_0(s))=\{\frac{1}{n}:n\in\mathbb{N}\}$ and, when compact, $\sigma(C_t,c_0(s))=\{\frac{1}{n}:n\in\mathbb{N}\}\cup\{0\}$. The results also connect to Goldberg classifications, providing a detailed partition of the spectrum for $R_a$ on weighted spaces and clarifying the interplay between the operator structure and the underlying weight sequences. Overall, the work extends classical Cesàro operator theory to weighted sequence spaces with explicit spectral descriptions and approximations.
Abstract
In this article, we conduct a comprehensive study on the continuity, compactness, and spectral properties of Rhaly operators and generalized Cesàro operators, acting on weighted null sequence spaces. We determine the point spectrum, continuous spectrum, and residual spectrum for compact Rhaly operators and compact generalized Cesàro operators. Additionally, we explore Goldberg's classifications of Rhaly operators over weighted null sequence spaces.