Vol. 18, No. 1, 1966

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Operators of Riesz type

Selwyn Ross Caradus

Vol. 18 (1966), No. 1, 61–71

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 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 . Next, perturbation theory is considered and it is shown that with suitable commutativity conditions, 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 .

Mathematical Subject Classification
Primary: 47.10
Secondary: 47.45
Received: 26 February 1965
Published: 1 July 1966
Selwyn Ross Caradus