The concept of an operator of
Riesz type was introduced by A. F. Ruston by using as an axiomatic system those
properties of compact operator used by F. Riesz in his original discussion of
integral equations. In this paper we first show that this system of axioms can
be somewhat simplified, and that in fact the class ℛ of operators of Riesz
type coincides with the class of bounded linear operators whose Fredholm
region consists of all nonzero complex numbers. It is further shown that the
class of strictly singular operators introduced by T. Kato and the class of
inessential operators introduced by D. C. Kleinecke both lie within ℛ. Next,
perturbation theory is considered and it is shown that with suitable commutativity
conditions, ℛ has the defining properties of a closed ideal. Finally, if f is
analytic on an open set containing σ(T) and f(0) = 0, then f(T) ∈ℛ if
T ∈ℛ. Moreover, if T ∈ℛ, then the algebra generated by T also lies within
ℛ.
