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

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

Published electronically:
May 29, 2002

MathSciNet review:
1933348

Full-text PDF

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,*Foundations of the theory of linear operators in spaces with an indefinite metric*, Nauka, Moscow, 1986 (Russian); English transl.:*Linear operators in spaces with an indefinite metric*, Wiley, New York, 1989. MR**90j:47042****[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. Brodskii,*On the properties of an operator mapping into itself the non-negative part of a space with an indefinite metric*, Uspekhi Mat. Naut**14**(1), 1959, 147-152 (Russian). MR**21:5145****[DR]**M.A. DRITSCHEL, J. ROVNYAK,*Extension theorems for contraction operators on Krein spaces*, Operator Theory: Adv. Appl., vol. 47, Birkhäuser Verlag, Basel, 1990, 221-305. MR**92m:47068****[KL]**M.G. KREIN AND H. LANGER,*On some mathematical principles in the linear theory of damped oscillations of continua*, Proc. Int. Sympos. on Applications of the Theory of Functions in Continuum Mechanics, Tbilisi, 1963, Vol. II: Fluid and Gas Mechanics, Math. Methods, Moscow, 1965, 283-322 (Russian); English transl.: Integral Equations Operator Theory**1**, 1978, 364-399 and 539-566. MR**80d:47026**; MR**80h:47022****[L1]**H. LANGER,*Zur Spektraltheorie -selbstadjungierter Operatoren*, Math. Ann.**146**(1), 1962, 60-85. MR**25:1450****[L2]**H. LANGER,*Eine Verallgemeinerung eines Satzes von L.S. Pontrjagin*, Math. Ann.**152**(5), 1963, 434-436. MR**28:1492****[LT]**H. LANGER AND CHR. TRETTER,*Spectral decomposition of some nonselfadjoint block operator matrices*, J. of Operator Theory**39**, 1998, 339-359. MR**99d:47004****[S]**A.A. SHKALIKOV,*On the existence of invariant subspaces of dissipative operators in an inner product space*, Fund. and Appl. Math.**5**(2), 1999, 627-635 (Russian). MR**2001k:47051****[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