Abstract
We study properties of Jordan representations ofH-dissipative operators in a finite-dimensional indefiniteH-space. An algebraic proof is given of the fact that such operators always have maximal semidefinite invariant subspaces.
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References
T. Ya. Azizov and I. S. Iokhvidov,Fundamentals of the Theory of Linear Operators in Spaces with Indefinite Metric [in Russian], Nauka, Moscow (1986).
A. I. Mal'tsev,Fundamentals of Linear Algebra, [in Russian], Gostekhizdat, Moscow-Leningrad (1948).
I. Gohberg, P. Lancaster, and L. Rodman,Matrices and Indefinite Scalar Products, Birkhäuser, Basel (1983).
A. A. Shkalikov, “Operator pencils arising in elasticity and hydrodynamics: the instability index formula,” in:Operator Theory: Advances and Applications, Vol. 87, Birkhäuser, Basel (1996), pp. 258–285.
T. Ya. Azizov and L. I. Sukhocheva, “A new approach to the proof of the theorem on the reduction of matrices to the Jordan normal form,” in:Functional Analysis. Linear Spaces [in Russian], Ulyanovsk (1990), pp. 3–5.
A. G. Kostyuchenko and A. A. Shkalikov, “Self-adjoint operator pencils and elliptic problems,”Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.],17, No. 2, 38–61 (1983).
A. A. Shkalikov, “Selection principles and properties of a part of root elements of operator pencils,”Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.], No. 4, 16–25 (1988).
A. A. Shkalikov, “Elliptic equations in Hilbert space and related spectral problems,”Trudy Seminara im. Petrovskogo,14, 140–224 (1989).
T. Ya. Azizov, P. A. Binding, J. Bognar, and B. Najman, “Nondegenerate Jordan subspaces of self-adjoint operators in indefinite spaces,”Linear Algebra Appl.,207, 37–48 (1994).
T. Ya. Azizov, “Invariant subspaces and criteria for the completeness of systems of root vectors ofJ-dissipative operators in the Pontryagin space ∏κ,”Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.],200, No. 5, 1015–1017 (1971).
M. G. Krein and G. Langer, “Definite subspaces and generalized resolvents of an Hermitian operator in the space ∏κ,”Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.],5, No. 7, 59–71 (1971).
A. C. M. Ran and D. Temme, “Dissipative matrices and invariant maximal semidefinite subspaces,”Linear Algebra Appl.,212/213, 169–213 (1994).
T. Ya. Azizov and H. Langer, “Some spectral properties of contractive and expansive operators in indefinite inner product spaces,”Math. Nachr.,162, 247–259 (1993).
T. Ya, Azizov, “Dissipative operators in Hilbert space with indefinite metric,”Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.],37, No. 3, 639–662 (1973).
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Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 163–169, February, 1998.
The authors axe grateful to Professor A. A. Shkalikov for useful discussions.
The research of the first author was supported by INTAS under grant No. 93-0249. The research of the second author was supported by the International Science Foundation and the Russian Government under grant No. NZP300.
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Azizov, T.Y., Barsukov, A.I. The algebraic structure ofH-dissipative operators in a finite-dimensional space. Math Notes 63, 145–149 (1998). https://doi.org/10.1007/BF02308753
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DOI: https://doi.org/10.1007/BF02308753