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On the boundedness of Hamiltonian operators. (English) Zbl 1009.47022

Summary: We show that a non-negative Hamiltonian operator whose domain contains a maximal uniformly positive subspace is bounded.

MSC:

47B50 Linear operators on spaces with an indefinite metric
46C20 Spaces with indefinite inner product (Kreĭn spaces, Pontryagin spaces, etc.)
47B44 Linear accretive operators, dissipative operators, etc.
47B25 Linear symmetric and selfadjoint operators (unbounded)
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[1] T. Ya. Azizov and I. S. Iokhvidov, Linear operators in spaces with an indefinite metric, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1989. Translated from the Russian by E. R. Dawson; A Wiley-Interscience Publication. · Zbl 0714.47028
[2] T.YA. AZIZOV, V.K. KIRIAKIDI, AND G.A. KURINA, An indefinite approach to the reduction of a non-negative Hamiltonian operator function to a block diagonal form, Funct. Anal. and Appl., 35 (3), 2001, 73-75 (Russian). · Zbl 1017.47011
[3] M. L. Brodskiĭ, On properties of an operator mapping the non-negative part of a space with indefinite metric into itself, Uspehi Mat. Nauk 14 (1959), no. 1 (85), 147 – 152 (Russian).
[4] Michael A. Dritschel and James Rovnyak, Extension theorems for contraction operators on Kreĭn spaces, Extension and interpolation of linear operators and matrix functions, Oper. Theory Adv. Appl., vol. 47, Birkhäuser, Basel, 1990, pp. 221 – 305. · Zbl 0727.47018
[5] M. G. Kreĭn and H. Langer, On some mathematical principles in the linear theory of damped oscillations of continua. I, Integral Equations Operator Theory 1 (1978), no. 3, 364 – 399. Translated from the Russian by R. Troelstra. , https://doi.org/10.1007/BF01682844 M. G. Kreĭn and H. Langer, On some mathematical principles in the linear theory of damped oscillations of continua. II, Integral Equations Operator Theory 1 (1978), no. 4, 539 – 566. Translated from the Russian by R. Troelstra. · Zbl 0401.47018
[6] Heinz Langer, Zur Spektraltheorie \?-selbstadjungierter Operatoren, Math. Ann. 146 (1962), 60 – 85 (German). · Zbl 0119.32003
[7] Heinz Langer, Eine Verallgemeinerung eines Satzes von L. S. Pontrjagin, Math. Ann. 152 (1963), 434 – 436 (German). · Zbl 0114.31604
[8] Heinz Langer and Christiane Tretter, Spectral decomposition of some nonselfadjoint block operator matrices, J. Operator Theory 39 (1998), no. 2, 339 – 359. · Zbl 0996.47006
[9] A. A. Shkalikov, On the existence of invariant subspaces of dissipative operators in a space with an indefinite metric, Fundam. Prikl. Mat. 5 (1999), no. 2, 627 – 635 (Russian, with English and Russian summaries). · Zbl 0960.47020
[10] YU.L. SHMUL’YAN, Division in the class of \(J\)-expansive operators, Mat. Sb. (N.S.) 74 (116), 1967, 516-525 (Russian); English transl.: Math. USSR-Sbornik 3, 1967, 471-479.
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