Some remarks on complex powers of (-\(\Delta\) ) and UMD spaces (Q916065)

If \(\Delta\) denotes the Laplacian operator and X a Banach space, we prove that if \((-\Delta)^{is}\otimes Id_ x\) is a bounded operator on \(L^ 2({\mathbb{R}};x)\) for all \(s\in {\mathbb{R}}\), then X is a UMD space.