tailieunhanh - Báo cáo toán học: "Compact perturbations of definitizable operators "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Nhiễu loạn nhỏ gọn của các nhà khai thác definitizable. | Copyright by INCREST 1979 J. OPERATOR THEORY 2 1979 63 - 77 COMPACT PERTURBATIONS OF DEFIN1TIZABLE OPERATORS p. JONAS and H. LANGER Let 2 e be a Krein space 1 with the indefinite scalar product X . The J-selfadjoint operator A in with dense domain A is called definitizable if there exists a polynomial p such that p x x 2 0 for all X 6 Q An where n denotes the degree of p1 . The non-real part ff0 A of the spectrum of a definitizable operator A with p A 0 consists of no more than a finite number of points which lie symmetrically with respect to the real axis see 8 5 . We denote the Riesz-Dunford projector corresponding to ƠO Ạ by Eo- A definitizable operator A with p A 0 has a spectral function see 8 1 here we use the notation and results of 5 . That is we have a finite possibly empty set c X R R u oo with the following property If X denotes the Boolean algebra of subsets of R generated by the closed and open intervals whose endpoints do not belong to c A there exists a homomorphism E from X into a Boolean algebra of J-selfadjoint projectors in such that for A e 23Ộ4 we have 1 R I Eo- 2 AE A E A A 3 ff A d cd 4 16 c A if and only if the subspace E A w is indefinite1 2 f or all A containing t. The elements of c A are called critical points of A. If t e c A t oo then see . 1 we have pit 0 for each definitizing polynomial p of A. 1 In 1 J-selfadjoint operators are called selfadjoint and instead of definitizable the term positizable is used. 2 We recall that an element X e .yf is said to be positive non-negative neutral etc. if x x 0 x x 2 0 x x 0 etc. A subspace of is called positive non-negative neutral etc. if all its non-zero elements have this property it is called indefinite if it contains positive and negative elements. Furthermore we set ÍỊ5 x x 0 ipo 5p n 64 p. JONAS and H. LANGER For t e R we denote by x t A x_ t A the minimum of the numbers3 y E A yd x_ F d j where A runs over all open A e 4 with t e A and put x t A min x f A x_ t A . This quantity is