About the quotient of two bounded operators

Abstract

The quotient operators are introduced in order to extend the class of all closed operators acting on a Hilbert space H. In fact, Kaufman proved in Kaufman (Proc Am Math Soc 72:531-534, 1978) using A and B such that (Formula Presented.) is closed in H, that a linear operator T on H is closed if and only if T is represented as a quotient B/A. So that every closed operator is included in the class of quotients. Moreover, he proved that if T is a closed densely defined operator, then T is represented as (Formula Presented.) using a unique pure contraction B, i.e., an operator such that (Formula Presented.) for all nonzero x in H. In this paper we attempt to study some algebraic and topological properties of quotient operators acting on Hilbert space, such that the boundedness, compactness and invertibility, other results such that the powers of quotient and the quotient character of limit of converging sequence of quotient operators are also established.