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

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 $\mid \mathrm{I}\mathrm{m}\lambda \mid \le 2\parallel T{P}^{-}\parallel$. We prove that in both half-planes $\mid \mathrm{I}\mathrm{m}\lambda \mid$ never exceeds $t_0\parallel T{P}^{-}\parallel$ 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.