Characterization of UMD Banach spaces by imaginary powers of Hermite and Laguerre operators (Q371841)

The authors characterize \(UMD\) spaces by means of the \(L^p\)-boundedness properties of the imaginary powers of the Hermite and Laguerre operators. More specifically, considered are natural self-adjoint extensions of either the Hermite operator (harmonic oscillator) \(-\frac12\big(\frac{d^2}{dx^2}-x^2\big)\) acting on functions on \(\mathbb R\), or the Laguerre operator \(-\frac12\big(\frac{d^2}{dx^2}-x^2-\frac{\alpha^2-1/4}{x^2}\big)\), \(\alpha>-1/2\), acting on functions on \(\mathbb R_+\). If \(\mathcal L\) denotes one of these two operators and \(\Omega=\mathbb R\) (Hermite case) or \(\Omega=\mathbb R_+\) (Laguerre case), then it is proved that a Banach space \(B\) has the \(UMD\) property if and only if for a \(p\), \(1<p<\infty\) (equivalently, for any such \(p\)), the imaginary powers \(\mathcal L^{i\gamma}\), \(\gamma\in \mathbb R\), defined on \(L^p_B(\Omega,dx)\otimes B\), can be extended, as bounded operators, to the Bochner-Lebesgue space \(L^p_B(\Omega,dx)\).