Guerre-Delabriere, Sylvie Some remarks on complex powers of (-\(\Delta\) ) and UMD spaces. (English) Zbl 0703.47024 Ill. J. Math. 35, No. 3, 401-407 (1991). 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. Reviewer: S.Guerre-Delabriere Cited in 1 ReviewCited in 11 Documents MSC: 47B38 Linear operators on function spaces (general) 60G46 Martingales and classical analysis 46E40 Spaces of vector- and operator-valued functions 47A50 Equations and inequalities involving linear operators, with vector unknowns 47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.) 47F05 General theory of partial differential operators Keywords:complex powers of positive operators; vector-valued martingales; Laplacian operator; bounded operator; UMD space PDFBibTeX XMLCite \textit{S. Guerre-Delabriere}, Ill. J. Math. 35, No. 3, 401--407 (1991; Zbl 0703.47024)