On analyticity of semigroups on Bochner spaces and on vector-valued noncommutative ${\mathrm{L}}^p$-spaces

Abstract

We show that the analyticity of semigroups $(T_t{)}_{t\ge 0}$ of (not necessarily positive) selfadjoint contractive Fourier multipliers on ${\mathrm{L}}^p$-spaces of any abelian locally compact group is preserved by the tensorisation of the identity operator ${\mathrm{Id}}_X$ of a Banach space $X$ for a large class of K-convex Banach spaces, answering partially a conjecture of Pisier. The result is even new for semigroups of Fourier multipliers acting on ${\mathrm{L}}^p({\mathbb{R}}^n)$. The proof relies on the use of noncommutative Banach spaces and we give a more general result for semigroups of Fourier multipliers acting on noncommutative ${\mathrm{L}}^p$-spaces. Finally, we also give a somewhat different version of this result in the discrete case, i.e. for Ritt operators.

21 pages, extension of the results to amenable groups