Spectral estimates and basis properties for self-adjoint block operator matrices

TL;DR

This paper explores the spectral properties of self-adjoint block operator matrices, establishing relationships with their diagonal entries, and investigates basis properties of eigenvectors, with applications to magnetohydrodynamics.

Contribution

It introduces new spectral enclosures and basis criteria for eigenvectors of self-adjoint operator matrices, linking spectral analysis with graph invariants and angular operators.

Findings

Spectral enclosures for eigenvalues derived from diagonal entries.

Existence of bounded angular operators implies basis properties.

Application to magnetohydrodynamics demonstrates practical relevance.

Abstract

In the first part of this manuscript a relationship between the spectrum of self-adjoint operator matrices and the spectra of their diagonal entries is found. This leads to enclosures for spectral points and in particular, enclosures for eigenvalues. We also consider graph invariant subspaces, and their corresponding angular operators. The existence of a bounded angular operator leads to basis properties of the first component of eigenvectors of operator matrices for which the corresponding eigenvalues lie in a half line. The results are applied to an example from magnetohydrodynamics.