It is a classical result that, if T is a maximal symmetric operator in a Krein space \(\mathscr {H}=\mathscr {H}^+[\dotplus ]\mathscr {H}^-\) with the domain \(\mathscr {D}_T\supset \mathscr {H}^-\) , then the imaginary part of its eigenvalue \(\lambda \) from upper or lower half-plane is bounded by \(|{{\,\textrm{Im}\,}}\lambda |\leqslant 2\Vert TP^-\Vert \) . We prove that in both half-planes \(|{{\,\textrm{Im}\,}}\lambda |\) never exceeds \(t_0\Vert TP^-\Vert \) 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.
