Abstract
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 local and even yields inequalities between gradations of K-convexity norms and Walsh type norms of operators. Our method combines G. Pisier's original ideas and the main steps in the proof of the Beurling-Kato theorem on extensions of C0-semigroups of operators to holomorphic semigroups with the technique of ideal norms.
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