Geodesic
A Note on the Caradus Class of BoundedLinear Operators on a Complex Banach Space
Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 592-594

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1. In a recent paper (1) on meromorphic operators, Caradus introduced the class of bounded linear operators on a complex Banach space X. A bounded linear operator T is put in the class if and only if its spectrum consists of a finite number of poles of the resolvent of T. Equivalently, T is in if and only if it has a rational resolvent (8, p. 314).Some ten years ago (in May, 1957), I discovered a property of the class g which may be of interest in connection with Caradus' work, and is the subject of the present note. 2. THEOREM. Let X be a complex Banach space. If T belongs to the class , and the linear operator S commutes with every bounded linear operator which commutes with T, then there is a polynomial p such that S = p(T). 
Ruston, A. F. A Note on the Caradus Class of BoundedLinear Operators on a Complex Banach Space. Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 592-594. doi: 10.4153/CJM-1969-066-6
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