Scalar-type spectral operators and holomorphic semigroups

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