On operators of transition in Krein spaces

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.