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Báo cáo toán học: "A projection-property for abstract rational (1-point) approximants "
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Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một tài sản chiếu cho hợp lý trừu tượng (1 điểm) approximants. | . OPERATOR THEORY 10 1983 127-131 Copyright by JNCREST 1983 A PROJECTION-PROPERTY FOR ABSTRACT RATIONAL 1-POINT APPROXIMANTS ANNIE A. M. CUYT 1. NOTATION AND DEFINITIONS Consider the operator F X - Y analytic in 0 2 pp. 113 where Xt 0 is a Banach space and Y 0 1 is a commutative Banach algebra without nilpotent elements 0 is the unit for addition and I is the unit for multiplication . The scalar field is R or c. A nonlinear operator P X- Y such that P x A x . Ao with A y. X t Ya symmetric and bounded z-linear operator z 0 . n is called an abstract polynomial 2 pp. Ill The degree of P x is n. The notation for the exact degree of P x is ÕP the largest integer k with Akxk 0 and the notation for the order of P x is dop the smallest integer k with Akxk 0 . Write D F - x e XI F x is regular in Y i.e. there exists ye Y F x -y Since F is analytic in 0 there exists r 0 such that F x -L F fc 0 xfc for x r. fc o k We say that F x O x for j eN if there exist J 6 Ro and 0 r 1 such that liF x J x Definition 1.1. The couple of abstract polynomials Ax g x - Anm nx . Alimx Bnm mxnni m . B xnm is called a solution of the Fade approximation problem of order n m for F if the abstract power series F-Q - P x O x 1 . 128 ANNIE A. M. CUYT We define the operator--- D Q Y by x -- ộ x -1 the inverse element of Q x for the multiplication in Y. We call the abstract rational operator 1 . p the quotient of two abstract polynomials reducible if there exist abstract polynomials T R and s such that p T-R Q -- TS and cP 1. Let us assume that the Banach space X and the Banach algebra Y are such that the irreducible form of an abstract rational operator is unique and that the abstract rational approximant of order n m for F see Definition 1.2 is unique. The matter was discussed in 1 . Definition 1.2. Let P Q be a couple of abstract polynomials satisfying Definition 1.1 with D P uD 2 0. The irreducible form - J- PS of 1 Pis ớ Q called the abstract rational approximant of order n m for F abbreviated n m