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