Geodesic
Invariant Subspaces of Dissipative Operators in a~Space with Indefinite Metric
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Studies on function theory and differential equations, Tome 248 (2005), pp. 294-303

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A theorem on the existence of maximal nonnegative invariant subspaces is proved for a special class of dissipative operators in a Hilbert space with indefinite inner product. It is shown that the spectra of the restrictions of these operators on the corresponding invariant subspaces lie in the closed upper half-plane. The theorem obtained is a generalization of the well-known results of L. S. Pontryagin, H. K. Langer, M. G. Krein, and T. Ya. Azizov devoted to this subject. 
@article{TM_2005_248_a26,
     author = {A. A. Shkalikov},
     title = {Invariant {Subspaces} of {Dissipative} {Operators} in {a~Space} with {Indefinite} {Metric}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {294--303},
     publisher = {mathdoc},
     volume = {248},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2005_248_a26/}
}
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A. A. Shkalikov. Invariant Subspaces of Dissipative Operators in a~Space with Indefinite Metric. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Studies on function theory and differential equations, Tome 248 (2005), pp. 294-303. http://geodesic.mathdoc.fr/item/TM_2005_248_a26/