On Factorizations of Analytic Operator-Valued Functions and Eigenvalue Multiplicity Questions

TL;DR

This paper investigates eigenvalue multiplicities in non-self-adjoint operators using factorizations of analytic operator-valued functions, extending existing results and connecting algebraic multiplicities with operator indices.

Contribution

It introduces new techniques based on factorizations to analyze eigenvalue multiplicities and extends the Weinstein-Aronszajn formula to broader operator classes.

Findings

Re-proved and extended results by Latushkin and Sukhtyaev.

Established a relation between algebraic multiplicities and operator indices.

Connected the Fredholm index of projection pairs to the Birman-Schwinger operator.

Abstract

We study several natural multiplicity questions that arise in the context of the Birman-Schwinger principle applied to non-self-adjoint operators. In particular, we re-prove (and extend) a recent result by Latushkin and Sukhtyaev by employing a different technique based on factorizations of analytic operator-valued functions due to Howland. Factorizations of analytic operator-valued functions are of particular interest in themselves and again we re-derive Howland's results and subsequently extend them. Considering algebraic multiplicities of finitely meromorphic operator-valued functions, we recall the notion of the index of a finitely meromorphic operator-valued function and use that to prove an analog of the well-known Weinstein-Aronszajn formula relating algebraic multiplicities of the underlying unperturbed and perturbed operators. Finally, we consider pairs of projections for which the difference belongs to the trace class and relate their Fredholm index to the index of the naturally underlying Birman-Schwinger operator.