tailieunhanh - Báo cáo toán học: "Continuous and analytic invariants for deformations of Fredholm complexes "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Bất biến liên tục và phân tích biến dạng của khu phức hợp Fredholm. | Copyright by INCREST 1983 J. OPERATOR THEORY 9 1983 3 - 26 CONTINUOUS AND ANALYTIC INVARIANTS FOR DEFORMATIONS OF FREDHOLM COMPLEXES M. PUTINAR and . VASILESCU 1. INTRODUCTION The aim of this paper is to prove the norm continuity of certain invariants that are attached to Fredholm complexes of Banach spaces and their endomorphisms. In particular we prove the continuity of the Lefschetz number which can naturally be defined in this context. The analytic and smooth dependence of these invariants will also be considered. The usual stability of the index of a Fredholm operator under small perturbations can of course be regarded as a continuity statement. Starting from this simple remark the second named author has proved in 14 the norm continuity of the Lefschetz number attached to a Fredholm operator and a pair of operators that intertwines it as a function of three arguments. The first named author has then noticed that similar results can be proved for a larger class of invariants that is derived from the characteristic polynomial. In this paper we shall extend these considerations tó the case of Fredholm complexes of Banach spaces. Roughly speaking for each pair consisting of a Fredholm complex and an endomorphism of it we define a rational function which is called here the characteristic function see Definition below and show that this assignment is norm continuous in a neighbourhood of the origin in the complex plane. In particular the coefficients of the Taylor expansion of the characteristic function at the origin are norm continuous complex-valued functions. In order to state more accurately the main result Theorem below let US introduce some notations and definitions. Let X and Y be Banach spaces over the complex field C and let US denote by y X y the space of all continuous linear operators from X into Y. The space YfX X will be simply denoted by .Y X . For every Se X X Y we denote by N S R S and y S the null-space the range and the reduced .