Homogeneous operators and essential complexes

Abstract

1. 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. We obtain some generalized homotopic properties of the class of essential Fredholm complexes, in our sense, which are then applied to establish its relationship with similar concepts. We also prove the stability of this class under small perturbations with respect to the gap topology.Throughout this paper we shall work with linear spaces over the field K, which is either the real field U or the complex one C. Let Ban K be the category whose objects are Banach spaces over IK and whose morphisms are bounded IK-linear operators. In the present work a complex in the category Ban K is a sequence A = (A p ) p^0 , where A p e i£{X p , Xp^), A p A p+1 = 0 and X p is a Banach space over IK for every integer p 3= 0, with A

Topologies on the Set of All Subspaces of a Banach Space and Related Questions of Banach Space Geometry

“…The geometric opening and quantity Θ 0 turned out to be important tool in the creation of the theory of Fredholm operators in Banach spaces [GK], [K1]. They are repeatedly used in the theory of operators in Banach spaces (see, in particular, [AZ], [CPY], [GM2], [Go], [GoM], [MeS], [Sob], [Va2], [Va3]), Fredholm and semi-Fredholm complexes of Banach spaces (see [AV], [A], [Do], [F], [FS], [Va1]).…”