Suppose that $f$ is a function on $\mathbb{R}^n$ such that
$\exp(a |\cdot|^2) f$ and $\exp(b |\cdot|^2) \hat f$ are bounded, where
$a,b > 0$. Hardy's Uncertainty Principle asserts that if $ab > \pi^2$,
then $f = 0$, while if $ab = \pi^2$, then $f = c\exp(-a|\cdot|^2)$.
In this paper, we generalise this uncertainty principle to vector-valued
functions, and hence to operators. The principle for operators can be
formulated loosely by saying that the kernel of an operator cannot be
localised near the diagonal if the spectrum is also localised.