Spectral analysis for finite rank perturbations of diagonal operators in non-archimedean Hilbert space

Abstract

In this paper we are concerned with the spectral analysis for some classes of finite rank perturbations of diagonal operators in the form, A = D + F, where D is a diagonal operator and F = u1 ⊗ v1 + u2 ⊗ v2 + … + um ⊗ vm is an operator of finite rank in the non-archimedean Hilbert space \(\mathbb{E}_\omega \). Using the theory of Fredholm operators in the non-archimedean setting and the concept of essential spectrum for linear operators, we compute the spectrum of A. A few examples are given at the end of the paper to illustrate our main results.