Components of invertible elements in quotient algebras of operators

Abstract

The purpose of this article is to present certain aspects of the theory of invertible elements in the various quotient algebras of the algebra, Q(S), of all bounded operators on a Hilbert space SD. In particular, the maximal connected subsets (components) of the set of left or right invertible elements in these quotient algebras are classified. A preliminary step is the geometric characterization of the operators in Q(S) that are left or right invertible modulo an ideal of this algebra. If ID is a Hilbert space and a is any infinite cardinal not larger than the dimension of ID then we denote by I the set of operators in Q(S) with the property that their ranges contain no closed subspace of dimension a. It is part of the folklore that these sets . are the only closed two-sided ideals of 3(S). When a is the smallest infinite cardinal No it is known that 4 is the ideal of compact operators. (Lemma 1 contains this result.) The transformations from one Banach space to another that have left or right inverses modulo the compact operators were characterized by Yood [5, p. 609]. It follows from the work of Atkinson [1] on Fredholm operators that there are infinitely many components of operators invertible modulo the compact operators on a Banach space E, if there is a subspace F that is isomorphic to E and E/F is finite dimensional. The exact composition of these components was discovered by Cordes and Labrousse [4] for the case of a Hilbert space. In what follows we will also restrict our attention to Hilbert space but since we consider all ideals 4, not just the ideal of compact operators, our results properly include those described above; at least in the case of a nonseparable Hilbert space. The set of operators that are left invertible modulo 4 will be denoted by S+(a). To be perfectly explicit: