On operators of transition in Krein spaces

• Autorzy: Grod A. , Kuzhel S. , Sudilovskaya V.
• Języki publikacji: EN

Abstrakty

EN

The paper is devoted to investigation of operators of transition and the corresponding decompositions of Krein spaces. The obtained results are applied to the study of relationship between solutions of operator Riccati equations and properties of the associated operator matrix L. In this way, we complete the known result (see Theorem 5.2 in the paper of S. Albeverio, A. Motovilov, A. Skhalikov, Integral Equ. Oper. Theory 64 (2004), 455-486) and show the equivalence between the existence of a strong solution K (//K// < 1) of the Riccati equation and similarity of the J-self-adjoint operator L to a self-adjoint one.

Słowa kluczowe

EN

Krein spaces; indefinite metrics; operator of transition; operator Riccati equation

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2011
• Tom: Vol. 31, no. 1
• Strony: 49--59
• Opis fizyczny: Bibliogr. 13 poz.

Twórcy

autor

Grod A.

autor

Kuzhel S.

autor

Sudilovskaya V.

• Taras Shevchenko National University 03127, Kyiv, Ukraine,

Bibliografia

• [1] V. Adamyan, H. Langer, C. Tretter, Existence and uniqueness of contractive solutions of some Riccati equations, J. Funct. Analysis 179 (2001), 448-473.
• [2] S. Albeverio, U. Günther, S. Kuzhel, J-Self-adjoint operators with C-symmetries: Extension Theory Approach, J. Phys. A 42 (2009), 105205-105221.
• [3] S. Albeverio, S. Kuzhel, Pseudo-Hermiticity and theory of singular perturbations, Lett. Math. Phys. 67 (2004), 223-238.
• [4] S. Albeverio, A. Motovilov, A. Shkalikov, Bounds on variation of spectral subspaces under J-self-adjoint perturbations, Integr. Equ. Oper. Theory 64 (2009), 455-486.
• [5] S. Albeverio, A. Motovilov, C. Tretter, Bounds on the spectrum and reducing subspaces of J-self-adjoint operators, arXiv:0909.1211v1 [math.SP] 7 Sep 2009.
• [6] T.Ya. Azizov, I.S. Iokhvidov, Linear Operators in Spaces with Indefinite Metric. Wiley, Chichester, 1989.
• [7] C.M. Bender, Making sense of non-Hermitian Hamiltonians, Rep. Prog. Phys. 70 (2007), 947-1018.
• [8] P. Jonas, On a class of selfadjoint operators in Krein spaces and their compact perturbations, Integr. Equat. Oper. Th. 11 (1988), 351-384.
• [9] A. Kuzhel, S. Kuzhel, Regular Extensions of Hermitian Operators, VSP, Utrecht, 1998.
• [10] S. Pedersen, Anticommuting self-adjoint operators, J. Funct. Analysis 89 (1990), 428-443.
• [11] R.S. Phillips, The extension of dual subspaces invariant under an algebra, [in:] Proc. of the International Symposium on Linear Spaces (Jerusalem, 1960), pp. 366-398, Jerusalem Academic Press, 1961.
• [12] T. Tanaka, General aspects of PT-symmetric and P-self-adjoint quantum theory in a Krein space, J. Phys. A 39 (2006), 14175-14203.
• [13] C. Tretter, Spectral Theory of Block Operator Matrices and Applications, Imperial College Press, London, 2008.