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The purpose of this paper is to give two characterizations of scalar-type spectral operators.

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IntroductionThe notion of a well-bounded operator was introduced by Smart (9). The properties of well-bounded operators were further investigated by Ringrose (6, 7), Sills (8) and Berkson and Dowson (2). Berkson and Dowson have developed a more complete theory for the type (A) and type (B) well-bounded operators than is possible for the general well-bounded operator. Their work relies heavily on Sills' treatment of the Banach algebra structure of the second dual of the Banach algebra of absolutely continuous functions on a compact interval.The main result of this paper (Theorem 5) is the characterisation of a type (B) operator by means of the weak compactness of its .s/j-operational calculus (as in Theorem 4.2 of (2)) and the description of the operational calculus using Stieltjes integrals of a kind suggested by Krabbe (5). Our results are also stronger than those of Berkson and Dowson in that we show thê -operational calculus for a type (B) operator to be continuous on pointwise convergent nets of uniformly bounded variation.We are indebted to H. R. Dowson for much valuable advice and encouragement during the preparation of this paper.