The inversion of a class of linear operators

Abstract

Let Q L denote the set of all quasi-continuous number valued functions on a number interval [a, b] which vanish at a and are left continuous at each point of (a,b]. Every linear operator, Jzf, on Q L which is continuous relative to the sup norm topology for Q L has a unique representation of the form = \ b f(t)dL(t f s), feQ L , a ^ s ^ 6, where all integrals Ja are taken in the <7-mean Stieltjes sense, and L is a function on the square a S o ^ b, satisfying the conditions of Definition 1.2. This paper is concerned primarily with those linear operators, the P-operators, which are abstractions from that class of linear physical systems whose output signals at a given time do not depend on their input signals at a later time; and with a sub-family of the P-operators, the Pi-operators which include all stationary linear_ operators. The Poperators are the Volterra operators on Q L . Necessary conditions and sufficient conditions for a P-operator to have an inverse which is a P-operator are found; and a necessary and sufficient condition for a Pi-operator to have an inverse which is a P-operator is given in Theorem 3.1. In addition it is shown that if Sf is a Pi-operator and JS^" 1 is a P-operator then JS^"