Characterization Of Scalar Type Operators

Abstract

A closed densely defined operator H, on a Banach space X, whose spectrum is contained in JR and satisfies the growth condition II (Z-H)-l II:::; CII~~:l Vz E iJR for some a ~ 0 and C > 0 is of (00,00+1) type R In studying spectral theory, the main interest has been finding a criteria for an operator to be of scalar type. Different approaches have been used so far, for example, Kantorovitz established a criteria using boundedness of operators with real spectrum acting on a reflexive Banach space. He also characterized scalar type operators using semigroup theory where it is shown that a bounded operator H is a scalar type if and only if iH generates a definite group. The method uses Laplace transform and mainly applies to bounded operators. Thus there is need to extend this characterization to unbounded operators. If a > 0, fEU where U is an algebra of smooth functions and H is of (a, a + 1) type JR, then the integral f(H) := -~ Ie ~{(z - H)-ldxdy is norm convergent and defines an operator in B(X) with II f(H) II:::; c II f Iln+l' The map f --t f(H) is a U- functional calculus for H. Using this functional calculus and the semigroup of (a, 00+1) type JR operators, we characterize scalar type operators H satisfying the first inequality. We determine the necessary condition for a densely defined closed linear operator H acting on a Hilbert space 1-l to be of scalar type for f in the algebra of smooth functions U. We also characterize the scalar type operators using the semi group theory of (a, 00+ 1) type JR operators and then give some applications of scalar type operators in decomposability and abstract Cauchy problems with appropriate boundary conditions. This functional calculus is important since it applies to both bounded and unbounded operators.