Modified Cauchy kernels and functional calculus for operators on Banach space

In Banach space operators with a bounded H^∞ functional calculus, Cowling et al. provide some necessary and sufficient conditions for a type-ω operator to have a bounded H^∞ functional calculus. We provide an alternate development of some of their ideas using a modified Cauchy kernel which is L¹ with respect to the measure |dz|/|z|. The method is direct and has the advantage that no transforms of the functions are necessary.