Locus of non-real eigenvalues of a class of linear relations in a Krein space

TL;DR

This paper extends classical bounds on the imaginary parts of eigenvalues of maximal symmetric operators in Krein spaces, establishing a universal bound involving the operator norm and applicable to dissipative extensions.

Contribution

It introduces a new universal bound for the imaginary parts of eigenvalues of symmetric relations in Krein spaces, generalizing previous results.

Findings

Bound on imaginary parts of eigenvalues is approximately 1.84 times the norm of $TP^-$.

Result applies to closed symmetric relations and their dissipative extensions.

Improves classical bounds for eigenvalues in Krein space operators.

Abstract

It is a classical result that, if a maximal symmetric operator $T$ in a Krein space $\mathcal{H}=\mathcal{H}^-[\oplus]\mathcal{H}^+$ has the property $\mathcal{H}^-\subseteq\mathcal{D}_T$, then the imaginary part of its eigenvalue $\lambda$ from upper or lower half-plane is bounded by $\lvert \mathrm{Im}\,\lambda\rvert\leq2\lVert TP^- \rVert$. We prove that in both half-planes $\lvert \mathrm{Im}\,\lambda\rvert$ never exceeds $t_0\lVert TP^- \rVert$ for some constant $t_0\approx1.84$. The result applies to a closed symmetric relation $T$ and carries on a suitable, most notably dissipative, extension.