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, ℒ, on QL
which is continuous relative to the sup norm topology for QL has a unique
representation of the form ℒf(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 ≦≦ 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 ℒ is a P1-operator and ℒ−1 is a P-operator then ℒ−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.
