Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Máy lẻ của một số sản phẩm miễn phí C *- đại số. | J. OPERATOR THEORY 6 1981 143-153 Copyright by INCREST 1981 FREDHOLM OPERATORS AND THE CONTINUITY OF THE LEFSCHETZ NUMBER . VASILESCU 1. Let X be a complex Banach space and let X be the algebra of all linear bounded operators on X. if A e 2 X and M N are closed subspaces of X that are invariant under A such that N c M and dimAỈ ưV oo then we denote by Trjư N A the trace of the operator induced by A in M N. In particular if N 0 then di mA co and we write simply Trw 1 instead of TrA 0 X . Now consider another Banach space Y and let US denote by C X y the space of all bounded linear operators from X into y. If s e ÍỄ X Y is a Fredholm operator and A 6 3 X Beỉ Y have the property SA BS then we define LS Ấ B TrN S T TryyR S B where N S and R S stand for the null-space and the range of s respectively. The number will be called the Lefschetz number of the pair A B with respect to S see 1 for a similar context . The aim of this paper is to prove the norm-continuity of the function S A B - Ls A B when S runs over all Fredholm operators in f X Y and Ae C X Be Y satisfy SA BS. Note that for A 1 X and B 1Y we have the equality Lsơx ly inds therefore the property of continuity of the function Ls x B is more comprehensive than that of continuity . stability of the index for Fredholm operators. The last section of the present work contains some applications which are derived from the above mentioned continuity. In particular the existence of eigenvalues for some types of operators is proved. 144 . VASILESCU Now we shall state without proof two results concerning the trace of an operator that are more or less known. . Lemma. Let X and let M c X be a closed subspace that is invariant under A with dimX Af oo. Let N be a complement of M in X and let p denote the projection of X onto N along M. If AN PA I N then Trjv Xw Trx M A . . Lemma. Let A 6 XfX and let M N be closed subspaces of X that are invariant under A such that N cz M dimXIM oo and dimM N oo. Then
