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

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 $X$.