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Báo cáo toán học: "Stability of the index of a complex of Banach spaces "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Tính ổn định của các chỉ số của một phức tạp của không gian Banach. | J. OPERATOR THEORY 2 1979 247-275 Copyright by INCREST 1979 STABILITY OF THE INDEX OF A COMPLEX OF BANACH SPACES F.-H. VASILESCU 1. PRELIMINARIES Let X and Y be two Banach spaces over the complex field c. We denote by X Y the set of all linear and closed operators defined on linear submanifolds of X assigning values in Y. The subset of those operators of X Y which are everywhere defined hence continuous will be denoted by 3fiX Y . We write W and l X for V X X and ăfix X respectively. We put also X ăă X C i.e. the dual space of X. For every S 6 VIX Y we denote by Z S R S and N S the domain of definition the range and the null-space of s respectively. We recall that the index of s is given by 1.1 ind s dim N S - dim Y R S provided that R S is closed in Y and at least one of the numbers dim N S dim y 7 S is finite. For every complex vector space M we denote by dim M the algebraic dimension of M. If we represent the action of s by the sequence 1.2 0 X y- 0 not forgetting that s acts only on Z S y then the number 1.1 may be interpreted as the Euler characteristic of the complex 1.2 see 9 or 7 . This remark suggests a more general definition of the index which will be presented in the sequel. Consider a countable family of Banach spaces and a family of opera- tors a.p e cd Xp yp 1 such that R p.p c iv ap 1 for each integer p. We represent them by the sequence ap P 1 1.3 ------- xp Xp X ---------- and we say that 1.3 is a cochain- complex of Banach spaces. The sequence X a Xp can be associated with the cohomology sequence H X a 248 F.-H. VASILESCU H X a i i0O where HP X 0 N pp R cip 1 . Let US assume that dim HP X a 00 for every integer p and that dim Hpfx a 0 for all but a finite number of indices. Then we may define 1.4 ind X a - l p dim Hp X a . p co The number ind X a which may be interpreted as the Euler characteristic of the complex 1.3 will be called shortly the index of the complex X a . It is easy to imagine a trick which makes possible the reduction of .