On a conjecture of Pisier on the analyticity of semigroups

We show that the analyticity of semigroups of selfadjoint contractive Fourier multipliers on -spaces of compact abelian groups is preserved by the tensorisation of the identity operator of a Banach space for a large class of K-convex Banach spaces, answering partially a conjecture of Pisier. We also give versions of this result for some semigroups of Schur multipliers and Fourier multipliers on noncommutative -spaces. Finally, we give a precise description of semigroups of Schur multipliers to which the result of this paper can be applied.