We show that a linear operator (possibly unbounded), A, on a reflexive Banach space, X, is a scalar-type spectral operator, with non-negative spectrum, if and only if the following conditions hold.

1. A generates a uniformly bounded holomorphic semigroup {e−^zA}_Re(z)≥0.

2. If FN(s)≡∫N−Nsin(sr)reirAdr$F_N (s) \equiv \int_{ - N}^N {\tfrac{{\sin (sr)}}{r}} e^{irA} dr$ , then {‖F_N‖} _N=1 ^∞ is uniformly bounded on [0,∞) and, for all x in X, the sequence {F_N(s)x} _N=1 ^∞ converges pointwise on [0, ∞) to a vector-valued function of bounded variation.

The projection-valued measure, E, for A, may be constructed from the holomorphic semigroup {e^−zA}_Re(z)≥0 generated by A, as follows.

$$\frac{1}{2}(E\{ s\} )x + (E[0,s)) x = \mathop {\lim }\limits_{N \to \infty } \int_{ - N}^N {\frac{{\sin (sr)}}{r}} e^{irA} x\frac{{dr}}{\pi }$$ for any x in X.