Notes on Rank One Perturbed Resolvent. Perturbation of Isolated Eigenvalue

TL;DR

This paper provides a didactic analysis of how rank-one perturbations affect isolated eigenvalues of operators, exploring whether such eigenvalues persist and how their multiplicities change.

Contribution

It offers a clear explanation of the impact of rank-one perturbations on isolated eigenvalues, including conditions for their persistence and changes in multiplicity.

Findings

Eigenvalues may or may not persist under rank-one perturbations.

Multiplicity of eigenvalues can increase, decrease, or stay the same after perturbation.

Krein's formula is used to analyze the spectral changes.

Abstract

This paper is a didactic commentary (a transcription with variations) to the paper of S.R. Foguel {\it Finite Dimensional Perturbations in Banach Spaces}. Addressed, mainly: postgraduates and related readers. Subject: Suppose we have two linear operators, A, B, so that B - A is rank one. Let \lambda_o be an {\it isolated} point of the spectrum of A. In addition, let \lambda_o be an {\it eigenvalue} of A: \lambda_o \in \sigma_{pp}(A) . The question is: Is \lambda_o an eigenvalue of B ? And, if so, is the multiplicity of \lambda_o in \sigma_{pp}(B) equal to the multiplicity of \lambda_o in \sigma_{pp}(A) ? -- or less? -- or greater? Keywords: M.G.Krein's Formula, Finite Rank Perturbation