Notes on Rank One Perturbed Resolvent. Perturbation of Isolated Eigenvalue
Abstract
This paper is a didactic commentary (a transcription with variations) to the paper of S.R. Foguel 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 λo be an isolated point of the spectrum of A. In addition, let λo be an eigenvalue of A: λo ∈ σpp(A) . The question is: Is λo an eigenvalue of B ? And, if so, is the multiplicity of λo in σpp(B) equal to the multiplicity of λo in σpp(A) ? -- or less? -- or greater? Keywords: M.G.Krein's Formula, Finite Rank Perturbation
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