Definitions. Throughout dK will denote a separable complex Hilbert space, and X will denote a complex Banach space. The set of bounded linear operators on X is denoted by B(X), and the set of all projections (that is, idempotent operators) on X by Proj(^f). Suppose that 9 is a Banach algebra of complex functions on some subset S c C , and that the functions e n (z) = z" lie in SF. Then by an 9 functional calculus we shall mean a Banach algebra homomorphism ip :&-+B (X) such that y>(e n ) = e n

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... T) = T", n = 0 , 1 , . . . . A contractive functional calculus is one for which \\ip\\ = 1. An operator T e B(X) is said to be scalar-type spectral if there is a spectral measure E defined on 53, the Borel subsets of C, with values in Proj(A r ) such that (i) E is countably additive on 53 in the strong operator topology; (ii) 7£(A) = £(A)7\ for all A e 53;