The Euler characteristic is stable under compact perturbations

Abstract

Abstract. We prove in the general case the stability under compact perturbations of the index (i.e. the Euler characteristic) of a Fredholm complex of Banach spaces. In particular, we obtain the corresponding stability property for Fredholm multioperators. These results are the consequence of a similar statement, concerning more general objects called Fredholm pairs.

Regular Fredholm Pairs

“…In [3] a Fredholm symmetrical pair was associated to each Fredholm complex. In fact, as above consider a complex of Banach spaces (X , d) and set…”

“…There are many ways to extend the notions of Fredholm operator and index to several variable operator theory. For instance, both Fredholm complexes of Banach spaces and the related notion of Fredholm pair have associated an index with good stability properties, see for example [9], [1], [2], [3].…”

Characterizations of Fredholm pairs and chains in Hilbert spaces

“…Since all the operators considered in this article will be defined on Hilbert spaces, all the definitions and facts reviewed will be restricted to this class of spaces and maps. For a general presentation, see the works [1], [2] and [6].…”

“…On the other hand, Fredholm pairs were studied in the works [1] and [2], where the main stability properties of such objects were also proved. Roughly speaking, the aforementioned pairs consist in an extension of the notion of Fredholm operator to multiparameter spectral theory, which is closely related to the concept of Fredholm Banach space complex, see [1] and [2]. However, Fredholm pairs have not been studied in the frame of Hilbert spaces yet.…”

Further results on regular Fredholm pairs and chains

“…On the other hand, Fredholm pairs and chains have been recently introduced and their main properties have been studied, see [1,2,4,5,8] and the monograph [3]. These objects consist in generalizations of the notions of Fredholm operators and Fredholm Banach space complexes respectively.…”