A remark on analytic Fredholm alternative

TL;DR

This paper explores the distribution of eigenvalues of Fredholm-type analytic operator-valued functions using a recent zero distribution result for analytic functions with a cut along the positive semi-axis.

Contribution

It applies a recent zero distribution theorem to analyze eigenvalues of Fredholm-type operator functions, providing new insights into their spectral properties.

Findings

Eigenvalues are constrained by Blaschke-type conditions.

The distribution of eigenvalues relates to the zeros of associated analytic functions.

The approach offers a new perspective on spectral analysis of operator-valued functions.

Abstract

We apply a recent result of Borichev-Golinskii-Kupin on the Blaschke-type conditions for zeros of analytic functions on the complex plane with a cut along the positive semi-axis to the problem of the eigenvalues distribution of the Fredholm-type analytic operator-valued functions.