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arXiv · 2412.15985

Keldysh's theorem revisited

Johannes M. Schumacher
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Abstract

In a variety of applications, the problem comes up of describing the principal part of the inverse of a holomorphic operator at an eigenvalue in terms of left and right root functions associated to the eigenvalue. Such a description was given by Keldysh in 1951. His theorem, the proof of which was published only in 1971, is a fundamental result in the local spectral theory of operator-valued functions. Here we present a streamlined derivation in the matrix case, and we extend Keldysh's theorem by means of a new principal part formula. Special attention is given to the semisimple case (first-order poles).

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Johannes M. Schumacher. 2024-12-20. Keldysh's theorem revisited. https://doi.org/10.1016/j.laa.2025.10.020

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