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We consider Galerkin approximations of holomorphic Fredholm operatoreigenvalue 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 notunconditionally satisfied and the question of convergence is delicate. Wereport a technique to prove regularity of approximations which is applicable toa wide range of eigenvalue problems. In particular, we introduce the conceptsof weak T-coercivity and T-compatibility and prove that for weakly T-coerciveoperators, T-compatibility of Galerkin approximations implies their regularity.Our framework immediately improves the results of [T. Hohage, L. Nannen, BIT55(1) (2015)], is immediately applicable to analyze approximations ofeigenvalue 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)].

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