The Closure of the Regular Operators in a Ring of Operators

1. Introduction. In [4]2 the elements of a Banach algebra are separated into various classes: the regular elements, the singular elements , the null divisors, the generalized null divisors, and so forth. These classes and their interrelations are then studied in detail. In [5] transformations between Banach spaces are studied and the classes of [4], after suitable generalization, are characterized. Among other things, it is proVed in [5] that each isomorphism between a Banach space and a proper, closed, linear subspace lies in the complement of the closure of the set of regular (invertible), continuous, linear transformations of the Banach space into itself relative to the norm (uniform) topology on the set of continuous, linear transformations. Precisely what transformations lie in the closure of the set of regular transformations seems nowhere to be recorded, even in the special case where the Banach space in question is a Hilbert space. This fact would seem to have some intrinsic interest as well as possible technical usefulness. In the following section, we describe completely those operators which lie in the closure of the set of regular operators in an arbitrary ring of operators (weakly closed, self-adjoint operator algebra on a Hilbert space)—cf. Theorem 1. Roughly speaking, this closure consists of those operators in the ring whose "essential domain" and