Modified Cauchy kernels and functional calculus for operators on Banach space

In Banach space operators with a bounded H-infinity functional calculus, Cowling et al. provide some necessary and sufficient conditions for a type-omega operator to have a bounded H-infinity functional calculus. We provide an alternate development of some of their ideas using a modified Cauchy kernel which is L-1 with respect to the measure \dz\/\z\. The method is direct and has the advantage that no transforms of the functions are necessary.