Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Khử cơ bản của các nhà khai thác selfadjoint trên không gian Krein. | J. OPERATOR THEORY 8 1982 219-225 Copyright by INCREST 1982 FUNDAMENTAL REDUC1BIL1TY OF SELFADJOINT OPERATORS ON KREĨN SPACE BRIAN w. McENNIS 1. INTRODUCTION An indefinite inner product space X with inner product is called a Krein space if there exists an operator J on such that J J J-1 and the J-inner product x y j - Zv y x Jy makes X a Hilbert space. Such an operator J is called a fundamental symmetry . it determines and is determined by the fundamental decomposition X_ where x Z-F J X and X_ I J X are Hilbert spaces with the inner products and respectively. See 3 Chapter V . The topology on X is that given by the J-norin II x j -- Jx x 1 2. A Krein space has many different fundamental symmetries but the J-norms obtained are all equivalent 3 Corollary Theorem . Throughout this paper A will denote a bounded linear operator on X. Unless otherwise stated concepts involving an inner product will be defined in terms of the indefinite inner product. Thus A is the operator satisfying Ax y x A y for all X y e X and A is selfadjoint if A A . If there is a fundamental decomposition that reduces A then A is called fundamentally reducible and the study of A can be reduced to the study of the Hilbert space operators A I X and A. I x_. It follows from 3 Lemma and Theorem VIII. that A is fundamentally reducible if and only if AJ J A for some fundamental symmetry J. Various conditions for an operator to be fundamentally reducible have been given some of which are referred to in the notes to Chapter VIII of 3 More recently Bajasgalan 2 has given a condition for the fundamental reducibility of a positive operator in terms of its spectral function. In Theorem 1 below we give necessary and sufficient conditions for the fundamental reducibility of a selfadjoint operator A. Conditions iii and iv of 220 BRIAN w. McENNIS Theorem 1 involve the following growth condition on the resolvent IỊM c Wi 1 .--1 2 3 . where M is a constant independent of k and i is a .
