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

LetT be a possibly unbounded linear operator in the Banach spaceX such thatR(t)=(t+T)^?1 is defined onR ⁺. LetS=TR(I?TR) and letB(.,.) denote the Beta function. Theorem 1.1.T is a scalar-type spectral operator with spectrum in [0, ∞) if and only if $$supleft{ {Bleft( {k,k} right)^{ - 1} int_0^infty {left| {x*S^k left( t right)x} right|{{dt} mathord{left/ {vphantom {{dt} t}} right. kern-nulldelimiterspace} t};left| x right| leqslant 1,} left| {x*} right| leqslant 1,k geqslant 1} right}< infty .$$ A “local” version of this result is formulated in Theorem 2.2.