Opuscula Math. 31, no. 1 (2011), 49-59
http://dx.doi.org/10.7494/OpMath.2011.31.1.49

Opuscula Mathematica

# On operators of transition in Krein spaces

A. Grod
S. Kuzhel
V. Sudilovskaya

Abstract. 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 $$K$$ ($$\|K\|\lt 1$$) of the Riccati equation and similarity of the $$J$$-self-adjoint operator $$L$$ to a self-adjoint one.

Keywords: Krein spaces, indefinite metrics, operator of transition, operator Riccati equation.

Mathematics Subject Classification: 47A55, 47A57, 47B25.

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• A. Grod
• Taras Shevchenko National University, 03127, Kyiv, Ukraine
• S. Kuzhel
• National Academy of Sciences of Ukraine, Institute of Mathematics, 01601, Kyiv, Ukraine
• AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Cracow, Poland
• V. Sudilovskaya
• National Pedagogical Dragomanov University, 01601, Kyiv, Ukraine
• Revised: 2010-07-03.
• Accepted: 2010-07-10.