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

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 $|\mathrm{Im}\lambda |\le 2\Vert T{P}^{-}\Vert$. We prove that in both half-planes $|\mathrm{Im}\lambda |$ never exceeds $t_0\Vert T{P}^{-}\Vert$ for some constant $t_0\approx 1.84$. The result applies to a closed symmetric relation $T$ and carries on a suitable, most notably dissipative, extension.