UMD Banach spaces and the maximal regularity for the square root of several operators

TL;DR

This paper establishes that the maximal L^p-regularity for the square root of certain differential operators on L^2 spaces characterizes the UMD property of the Banach space involved.

Contribution

It proves a characterization of the UMD property via maximal regularity for the square root of Hermite, Bessel, and Laguerre operators.

Findings

Maximal regularity characterizes UMD spaces.

Results apply to Hermite, Bessel, and Laguerre operators.

Provides new insights into operator theory on Banach spaces.

Abstract

In this paper we prove that the maximal $L^p$-regularity property on the interval $(0,T)$, $T>0$, for Cauchy problems associated with the square root of Hermite, Bessel or Laguerre type operators on $L^2(\Omega, d\mu; X),$ characterizes the UMD property for the Banach space $X$.