Vol. 19, No. 1, 1966

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The inversion of a class of linear operators

James Arthur Dyer

Vol. 19 (1966), No. 1, 57–66

Let QL 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, , on QL which is continuous relative to the sup norm topology for QL has a unique representation of the form f(s) = abf(t)dL(t,s), f QL, a s b, where all integrals are taken in the σ-mean Stieltjes sense, and L is a function on the square a ts 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 P1-operators which include all stationary linear operators. The P-operators are the Volterra operators on QL. 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 P1-operator to have an inverse which is a P-operator is given in Theorem 3.1. In addition it is shown that if is a P1-operator and 1 is a P-operator then 1 may be written as the product of two operators whose generating functions may be found by successive approximation techniques. An analogue of Lane’s inversion theorem for stationary operators on QCOL is found as a special case of these results.

Mathematical Subject Classification
Primary: 47.25
Secondary: 47.70
Received: 18 October 1965
Published: 1 October 1966
James Arthur Dyer