Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Pseudo-thường xuyên phổ chức năng trong không gian Krein. | Copyright by INCREST 1984 J OPERATOR THEORY 12 1984 349-358 PSEUDO-REGULAR SPECTRAL FUNCTIONS IN KREIN SPACES AURELIAN GHEONDEA 1. INTRODUCTION A pseudo-regular subspace is a subspace Ji of a Krein space X such that JI .y Ố where Jíữ is the isotropic part of JI and S9. is a regular subspace. If A is a definitizable operator then it is known 10 that A has an invariant maximal non-negative subspace. On the other hand the question when a definitizable operator has an invariant regular maximal non-negative subspace has a neat answer 12 Theorem 2 . This paper is concerned with the following question when a definitizable operator has an invariant pseudo-regular maximal non-negative subspace . Thus in Theorem we prove that a sufficient condition for the existence of an invariant pseudo-regular maximal non-negative subspace of the definitizable operator A is a certain condition of pseudo-regularity on the spectral function of A in the neighbourhoods of the critical points of A. However in Example it is shown that in general this pseudo-regularity condition is not necessary in order that A has an invariant pseudo-regular maximal non-negative subspace. Corollary gives a particular case in which the equivalence holds. We note that concerning this pseudo-regularity condition on the spectral function a slightly stronger condition appeared in 6 and also that the above considered problem can be related to 3 . The proof of the main result leans on Section 3 where the following problem is considered given a commutative family SP of selfadjoint projections decide when V PJf is a pseudo-regular subspace . We have presented in Section 1 some terminology from Krein space theory slightly different from some known papers according to what we need. The author expresses his gratitude to Gr. Arsene for valuable discussions on this subject and to p. Jonas for careful reading the manuscript and pertinent remarks. 350 AURELIAN GHEONDEA 2. REDUCTIONS BY NEUTRAL SUBSPACES Let X be a
