Spectral and Essential Spectral Analysis of Finite-Rank Perturbations of Unbounded Diagonal Operators on Non-Archimedean Hilbert Spaces

Abstract

We study the spectral properties of a class of unbounded linear operators on a non-Archimedean Hilbert space $E_{\omega }$. More precisely, we consider operators of the form \[ T=D+F,\qquad F=\sum_{j=1}^{m} u_j\otimes v_j, \] where $D$ is an unbounded diagonal operator and $F$ is a finite-rank perturbation. This work extends the spectral analysis of Diagana and McNeal for rank-one perturbations of diagonal operators to the case of arbitrary finite-rank perturbations. The main objective is to describe the spectrum, point spectrum, and essential spectrum of such operators in terms of the diagonal sequence associated with $D$ and the Fredholm properties of $\lambda I-T$. The theory of Fredholm operators plays a central role, particularly in the computation of the essential spectrum and in the study of stability under finite-rank perturbations.