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Keldysh's theorem revisited

Johannes M. Schumacher

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).

Keldysh's theorem revisited

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).

Paper Structure

This paper contains 8 sections, 33 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Partial multiplicities
  4. Partial multiplicities
  5. Root functions and canonical matrices
  6. Keldysh's theorem
  7. Extensions
  8. The semisimple case

Key Result

Theorem 3.1

There exist unimodular matrices $U_L, U_R \in \mathcal{H}_0^{n \times n}$ and uniquely determined nonnegative integers $m_1, \dots, m_n$ such that

Theorems & Definitions (72)

  • Theorem 3.1: Smith form w.r.t. $\mathcal{H}_0$
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • Proposition 3.4
  • proof
  • Example 3.5
  • Lemma 3.6
  • proof
  • Proposition 3.7
  • ...and 62 more