Vector-valued distributions and Hardy's uncertainty principle for operators

Suppose that f is a function on R-n such that exp(a vertical bar.vertical bar(2)) f and exp(b vertical bar.vertical bar(2)) (f) over cap 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 = cexp(-a vertical bar.vertical bar(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.