Analytic semigroups on vector valued noncommutative $L^p$-spaces

TL;DR

This paper establishes conditions under which operator semigroups on noncommutative L^p spaces, valued in operator spaces, are bounded analytic or R-analytic, and explores their functional calculus properties.

Contribution

It provides new sufficient conditions for analyticity of semigroups on vector valued noncommutative L^p spaces, extending previous results in the field.

Findings

Conditions for bounded analytic semigroups on noncommutative L^p spaces.

Extension of functional calculus results to vector valued noncommutative spaces.

Generalization of earlier work by Junge, Le Merdy, and Xu.

Abstract

We give sufficient conditions on an operator space $E$ and on a semigroup of operators on a von Neumann algebra $M$ to obtain a bounded analytic or a $R$-analytic semigroup $(T_t \otimes Id_E)_{t \geq 0}$ on the vector valued noncommutative $L^p$-space $L^p(M,E)$. Moreover, we give applications to the $H^\infty(\Sigma_\theta)$ functional calculus of the generators of these semigroups, generalizing some earlier work of M. Junge, C. Le Merdy and Q. Xu.