Equivalences compactes entre deux operateurs fermes sur un espace de Hilbert

Let H be a H ILBERT space, let C( H ) denote the set of all closed densely defined linear operators on H and let ( H ) denote the set of all bounded elements of C( H ). If A, B ( H ), set: A ∼ B iff A — B is compact. Then ∼ is an equivalence relation on ( H ) and many results have been stated and proved making use of this equivalence relation (e.g. correction of spectra by compact perturbations, C ALKIN algebras … etc …). The aim of the present paper is to show that it is possible to extend ∼ to the whole of C H ) in such a way that (after suitable modifications) most of the results referred to above on ( H ) are still true on C( H ).