Convergence of sequences of linear operators and their spectra

TL;DR

This paper investigates the spectral convergence of approximating sequences of unbounded non-selfadjoint operators with compact resolvents, providing conditions that guarantee accurate eigenvalue approximation without spurious results.

Contribution

It introduces general assumptions ensuring spectral exactness and establishes new conditions for strong convergence and discrete compactness of resolvent sequences.

Findings

Spectral convergence is achieved under generalized strong resolvent convergence.

Sufficient conditions for discrete compactness of resolvent sequences are provided.

Perturbation results for strong convergence are established.

Abstract

We establish spectral convergence results of approximations of unbounded non-selfadjoint linear operators with compact resolvents by operators that converge in generalized strong resolvent sense. The aim is to establish general assumptions that ensure spectral exactness, i.e. that every true eigenvalue is approximated and no spurious eigenvalues occur. A main ingredient is the discrete compactness of the sequence of resolvents of the approximating operators. We establish sufficient conditions and perturbation results for strong convergence and for discrete compactness of the resolvents.