# Galerkin approximation of holomorphic eigenvalue problems: weak T-coercivity and T-compatibility

## TL;DR

This paper develops a new framework using weak T-coercivity and T-compatibility concepts to analyze Galerkin approximations of holomorphic eigenvalue problems, ensuring their regularity and convergence.

## Contribution

It introduces the concepts of weak T-coercivity and T-compatibility, providing a general technique for proving regularity of Galerkin approximations in complex eigenvalue problems.

## Key findings

- Framework improves previous results on eigenvalue approximation.
- Applicable to a wide range of holomorphic Fredholm operator problems.
- Ensures convergence and regularity of Galerkin methods in non-coercive settings.

## Abstract

We consider Galerkin approximations of holomorphic Fredholm operator eigenvalue problems for which the operator values don't have the structure "coercive+compact". In this case the regularity (in sense of [O. Karma, Numer. Funct. Anal. Optim. 17 (1996)]) of Galerkin approximations is not unconditionally satisfied and the question of convergence is delicate. We report a technique to prove regularity of approximations which is applicable to a wide range of eigenvalue problems. In particular, we introduce the concepts of weak T-coercivity and T-compatibility and prove that for weakly T-coercive operators, T-compatibility of Galerkin approximations implies their regularity. Our framework immediately improves the results of [T. Hohage, L. Nannen, BIT 55(1) (2015)], is immediately applicable to analyze approximations of eigenvalue problems related to [A.-S. Bonnet-Ben Dhia, C. Carvalho, P. Ciarlet, Num. Math. 138(4) (2018)] and is already applied in [G. Unger, preprint (2017)].