Journal of the
Korean Mathematical Society JKMS

The Heisenberg inequality associated with the uncertainty principle is extended to an infinite dimensional abstract Wiener space $(H,B)$ with an abstract Wiener measure $p_1.$ For $\varphi \in L^2(p_1)$ and $T \in {\mathcal L}(B,H)$, it is shown that $$ \left[\int_B |Tx|_{_H}^{2}|\varphi(x)|^{2}p_1(dx)\right] \left[\int_B |Tx|_{_H}^{2}|{\mathcal F}\varphi(x)|^{2}p_1(dx)\right] \geq \|T|_{_H}\|_{_{HS}}^{4} \| \varphi \|_{2}^{4}, $$ where ${\mathcal F}\varphi$ is the Fourier-Wiener transform of $\varphi$. The conditions when the equality holds also discussed.

Keywords: abstract Wiener space, Fourier-Wiener transform, the Heisenberg uncertainty principle

MSC numbers: 46G12, 46N50, 46S50, 28C20