Change of Jordan structure of $G$-selfadjoint operators and selfadjoinl operator-functions under small perturbatios

The author considers the problem of the change of length of Jordan chains when passing from $G_0$-selfadjoint operator $A_0$ to $G$-selfadjoint operator $A$, provided $\|A-A_0\|+\|G-G_0\|$ is small enough. The role played by the so-called sign characteristics is clarified. The results will carry over to the case of small perturbations of holomorphic selfadjoint operator-valued functions.