SEMIGROUPS AND RESOLVENTS OF BOUNDED VARIATION, IMAGINARY POWERS AND H-INFINITY FUNCTIONAL-CALCULUS

Let -A be a linear, injective operator, on a Banach space X. We show that there exists an H(infinity) functional calculus for A if and only if -A generates a bounded strongly continuous holomorphic semigroup of uniform weak bounded variation, if and only if A(z + A)-1 is of uniform weak bounded variation. This provides a sufficient condition for the imaginary powers of A, {A(-is)}s is-an-element-of R, to extend to a strongly continuous group of bounded operators, we also give similar necessary conditions.