Characterization of unbounded spectral operators with spectrum in a half-line

Let T be a possibly unbounded linear operator in the Banach space X such that R(t)=(t+T)^-1 is defined on R ⁺. Let S=TR(I-TR) and let B(.,.) denote the Beta function. Theorem 1.1. T is a scalar-type spectral operator with spectrum in [0, ∞) if and only if {Mathematical expression} A "local" version of this result is formulated in Theorem 2.2.