Keldysh's theorem revisited
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909874266308608 |
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| author | Schumacher, Johannes M. |
| author_facet | Schumacher, Johannes M. |
| contents | 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). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_15985 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Keldysh's theorem revisited Schumacher, Johannes M. Functional Analysis 47A56, 15A54 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). |
| title | Keldysh's theorem revisited |
| topic | Functional Analysis 47A56, 15A54 |
| url | https://arxiv.org/abs/2412.15985 |