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Báo cáo toán học: "Some invariants for semi-Fredholm systems of essentially commuting operators "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một số bất biến cho các hệ thống bán Fredholm chủ yếu đi làm các nhà khai thác. | Copyright by INCREST 1982 J. OPERATOR THEORY 8 1982 65- 90 SOME INVARIANTS FOR SEMI-FREDHOLM SYSTEMS OF ESSENTIALLY COMMUTING OPERATORS MIHAI PUTINAR The aim of this paper is to give a sequence of integers associated in a natural way to a matrix Ty of essentially commuting operators on a Banach space which is semi-Fredholm and satisfies some condition e.g. there is a right or left essential inverse matrix Sji such that all the commutators Sji sk Sy TJ are compact . These numbers are invariant for compact or small norm perturbations of the Ty. s and we will study their properties. The invariants will be computed as indices of some Fredholm complexes associated with the matrix ST Ty . A Fredholm complex is an extension of the notion of complex of Banach spaces with finite dimensional cohomology and the index is a generalization of the Euler characteristic of such a complex. To do the construction of complexes we isolate the properties of .ỹ- in a universal framework Consider the polynomial ring A C Yyl with mn generators 1 i m 1 J n and consider also the map given by the matrix X Xy X An - Am. Tt turns out that the kernel of A has a finite free minimal resolution. This resolution specializes in our concrete situation to a Fredholm complex. The same procedure applied to some operators obtained from S by twisting with symmetric powers will give an entire sequence of Fredholm complexes associated with S whose indices will be denoted by indp 9 p 6 z. The first section contains the algebraic preliminaries namely the construction of the complexes and of certain homotopies. These complexes are obtained by mixing Koszul complexes corresponding to each line in the m X n-matrix with the symmetric algebra with m generators in a manner which reminds Spencer complexes 16 . Let us remark that one of these complexes appears in 4 related to some problems in algebraic dimension theory. 6-2159 66 MIHAI PCTINAR In 2 we use a notion of essential Fredholm complex of Banach spaces which