On the boundedness of Hamiltonian operators

Authors:
Tomas Ya. Azizov, Aad Dijksma and Irina V. Gridneva

Journal:
Proc. Amer. Math. Soc. **131** (2003), 563-576

MSC (2000):
Primary 47B50, 46C20, 47B44, 47B25

Published electronically:
May 29, 2002

MathSciNet review:
1933348

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

**[AI]**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. MR**1033489****[AKK]**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).**[B]**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). MR**0106413****[DR]**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. MR**1120277****[KL]**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. MR**511976**, 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. MR**516767**, 10.1007/BF01682940**[L1]**Heinz Langer,*Zur Spektraltheorie 𝐽-selbstadjungierter Operatoren*, Math. Ann.**146**(1962), 60–85 (German). MR**0138002****[L2]**Heinz Langer,*Eine Verallgemeinerung eines Satzes von L. S. Pontrjagin*, Math. Ann.**152**(1963), 434–436 (German). MR**0158266****[LT]**Heinz Langer and Christiane Tretter,*Spectral decomposition of some nonselfadjoint block operator matrices*, J. Operator Theory**39**(1998), no. 2, 339–359. MR**1620503****[S]**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). MR**1803604****[Shm]**YU.L. SHMUL'YAN,*Division in the class of -expansive operators*, Mat. Sb. (N.S.)**74**(116), 1967, 516-525 (Russian); English transl.: Math. USSR-Sbornik**3**, 1967, 471-479.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
47B50,
46C20,
47B44,
47B25

Retrieve articles in all journals with MSC (2000): 47B50, 46C20, 47B44, 47B25

Additional Information

**Tomas Ya. Azizov**

Affiliation:
Department of Mathematics, Voronezh State University, 394693 Voronezh, Russia

Email:
azizov@tom.vsu.ru

**Aad Dijksma**

Affiliation:
Department of Mathematics, University of Groningen, P.O. Box 800, 9700 AV Groningen, the Netherlands

Email:
dijksma@math.rug.nl

**Irina V. Gridneva**

Affiliation:
Department of Mathematics, Voronezh State University, 394693 Voronezh, Russia

DOI:
https://doi.org/10.1090/S0002-9939-02-06565-6

Received by editor(s):
March 13, 2001

Received by editor(s) in revised form:
September 28, 2001

Published electronically:
May 29, 2002

Additional Notes:
This research was supported by grants NWO 047-008-008 and RFBR 99-01-00391

Communicated by:
Joseph A. Ball

Article copyright:
© Copyright 2002
American Mathematical Society