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Báo cáo toán học: "Spectral shift functions that arise in perturbations of a positive operator. (Russian) "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Chức năng chuyển đổi quang phổ trong PHÁT SINH đó sự gián đoạn của một nhà điều hành tích cực. | J. OPERATOR THEORY 6 1981 . 195 -216 Copyright by INCREST 1981 ON THE PROPERTIES OF ABSTRACT RATIONAL 1-POINT APPROXIMANTS ANNIE A. M. CUYT 1. DEFINITION OF APA AND ARA Let F X - Y be an operator analytic in 0 5 pp. 113 where 0 is a Banach space and Y 2 0 1 is a commutative Banach algebra 0 is the unit for addition and I is the unit for multiplication X and Y have the same scalar field A where A is R or C . We always work in the norm-topology. Write D F x e x F x is regular in Y i.e. there exists J S Y F x -y I y-F xỳf For every positive integer k we consider the spaces L Xk Y LỊL is a -linear bounded operator L XL yw Y where L JV Y Y. So Lx . . . xk Lxfxv xk with Xj . . xt 6 xk and Lxk 6 LCL 1 y . The operator L e L Xk Y is called symmetric if Lxr. .xk Lx . .Xik for all x1 . . Xj G xk and all permutations z j .ik of 1 .k . An example of a symmetric -linear bounded operator is F A x0 the Á th Freeh Ct- derivative of Fin x0 5 pp. 100 110 . An abstract polynomial is a non-linear operator P X Y such that P x A xn . . Ao G Y with Al e L X Y and At symmetric. The degree of P x is n. We also introduce the following notations. If there exists a positive integer A such that Mk 0 k jk Akxk 0 and Ạ 0 then 30P j . If there exists a positive integer j2 such that k j2 k n Akxk 0 and Aj2xb f 0 then ÕP j2. ÕP is the exact degree of the abstract polynomial P cop is the order of the abstract polynomial p. We have the following important lemma for abstract polynomials. Lemma 1.1. Let u V be abstract polynomials X - Y. If U x -F x 0 and if D V 0 then u 0. The proof is given in 3 . 196 ANNIE A. M. CUYT Since F is analytic in 0 there exists an open ball 5 0 r with centre 0 e X and radius r 0 such that _ 1 F x s 77 FW 0 xk for IIXII r where i F ữ ỊS xữ 5 0 . We say that F x 0 j if there exist Je R f and an open ball 5 0 r with 0 r 1 such that x e 5 0 r 5 x J- x jeN . Write Ị 0 ck 6 L Xk Y . Definition 1.1. The couple of abstract polynomials P x Q x .A m nx m n . A mxnm 5 m rax . Bnmx m