On Characterization of Scalar Operators via Semigroup and the Associated Generators

Abstract

In this paper, we characterize the scalar operators by using the semigroup theory and the corresponding generators of (α, α + 1) type R operators. In particular, we show that if a densely defined operator H generates a contraction semigroup, then both H and H∗ are scalar operators and if H admits a U(Algebra of smooth functions) functional calculus of scalar type, then H∗ also admits a U functional calculus of scalar type.