Sergej A. Choroszavin ( sergius@pve.vsu.ru )
 Notes on Rank One Perturbed Resolvent.
 Perturbation of Isolated Eigenvalue.
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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.