K-convex operators and Walsh type norms

A celebrated result of G. PISIER states that the notions of B-convexity and K-convexity coincide for Banach spaces. We complement this in the setting of linear and bounded operators between Banach spaces. Our approach is loca l and even yields inequalities between gradations of K-convexity norms and Walsh type norms of operators. Our method combines G. PISIER'S original ide as and the main steps in the proof of the Beurling-Kato theorem on extensio ns of C-o-semigroups of operators to holomorphic semigroups with the techni que of ideal norms.