Stability of the index of a semi-Fredholm complex of Banach spaces

The Euler characteristic is stable under compact perturbations

“…It is known that if the sequence α = (α p ) p : 0→X [2]; see also [9] for a generalization). It was conjectured that in this case α and β have the same index, but no proof was known in general.…”

“…This is possible via some results of stability under small or compact perturbations, concerning Fredholm pairs (S, T ) (Definition 2). A particular case of this notion (namely for ST = 0, T S = 0) was introduced in [2]; see also [3], [4]. Most notation is similar to that in [2], [15].…”

Almost Exactness in Normed Spaces

“…When X,Y, and Z are Hilbert spaces then we can actually reduce the whole problem to this special case: observe [4,10] Thus the assumptions of Problem 3.1 are that the operator col(5, T*) : Y -* ZxX is bounded below, and can be approximated by the operators col(5n, T*) : Y -> ZxX which have dense range: then by Theorem 1.3 of [7] it follows that col{S,T*) is also dense, and hence almost open. If we strengthen the assumptions in Problem 3.1, assuming that the chains {Sn,Tn) form "short exact sequences", and use Theorem 2.3 to reduce to complete spaces, then an affirmative solution would seem to be contained in the indexstability results of Vasilescu and Albrecht [13,2] …”

Homogeneous operators and essential complexes

“…this case, one can define the index of A by the formula which has some stability properties (see [4], [14], [1], [2] etc. ).…”

“…Introduction. The aim of this work is to present a new approach to the concept of essential Fredholm complex of Banach spaces ([10], [2]; see also [11], [4], [6], [7] etc. for further connections), by using non-linear homogeneous mappings.…”