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

TL;DR

This paper proves that the analyticity of certain semigroups of Fourier multipliers on L^p-spaces is preserved under tensorization with a broad class of K-convex Banach spaces, extending to noncommutative L^p-spaces and Ritt operators.

Contribution

It establishes the preservation of analyticity for semigroups of Fourier multipliers on L^p-spaces under tensorization with K-convex Banach spaces, including noncommutative cases, partially confirming Pisier's conjecture.

Findings

Analyticity preserved for Fourier multiplier semigroups on L^p-spaces.

Extension of results to noncommutative L^p-spaces.

Application to Ritt operators in the discrete setting.

Abstract

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 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.