Geodesic
Change of Jordan structure of $G$-selfadjoint operators and selfadjoinl operator-functions under small perturbatios
Izvestiya. Mathematics , Tome 37 (1991) no. 2, pp. 371-395

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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. 
@article{IM2_1991_37_2_a5,
     author = {V. R. Ol'shevskii},
     title = {Change of {Jordan} structure of $G$-selfadjoint operators and selfadjoinl operator-functions under small perturbatios},
     journal = {Izvestiya. Mathematics },
     pages = {371--395},
     publisher = {mathdoc},
     volume = {37},
     number = {2},
     year = {1991},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1991_37_2_a5/}
}
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V. R. Ol'shevskii. Change of Jordan structure of $G$-selfadjoint operators and selfadjoinl operator-functions under small perturbatios. Izvestiya. Mathematics , Tome 37 (1991) no. 2, pp. 371-395. http://geodesic.mathdoc.fr/item/IM2_1991_37_2_a5/