Scalar-type spectral operators and holomorphic semigroups (Q1069109)

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 \(\{e^{- zA}\}_{Re(z)\geq 0}.\) (2) If \(F_ N(s)\equiv \int^{N}_{-N}\frac{\sin (sr)}{r}e^{irA}dr\), then \(\{\| F_ N\| \}^{\infty}_{N=1}\) is uniformly bounded on [0,\(\infty)\) and, for all x in X, the sequence \(\{F_ N(s)x\}^{\infty}_{N=1}\) converges pointwise on [0,\(\infty)\) to a vector-valued function of bounded variation. The projection-valued measure, E, for A, may be constructed from the holomorphic semigroup \(\{e^{-zA}\}_{Re(z)\geq 0}\) generated by A, as follows. \[ \frac{1}{2}(E\{s\})x+(E[0,s))x=\lim_{N\to \infty}\int^{N}_{-N}\frac{\sin (sr)}{r}e^{irA}x\frac{dr}{\pi}, \] for any x in X.