Operators, absolutely indefinitely bounded below, in spaces with indefinite metric

Let $r$ be the spectral radius of an operator $\mathfrak{U}$, absolutely indefinitely bounded below. It is proved that $r\geqslant c^{1/\alpha}$, where $c$ is the exact lower bound of $\mathfrak{U}$ and $\alpha$ is a number occurring in the definition of the $I$-metric. A bound is obtained for the dimensionality of the direct sum of root lineals of $\mathfrak{U}$ ($c\geqslant1$), corresponding to eigenvalues whose absolute values are smaller than unity.