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

TL;DR

This paper extends Hardy's uncertainty principle to vector-valued functions and operators, showing that operators with localized spectra cannot have kernels localized near the diagonal.

Contribution

It introduces a generalization of Hardy's uncertainty principle applicable to vector-valued functions and operators, linking kernel localization to spectral localization.

Findings

Operators with localized spectra cannot have kernels localized near the diagonal

Generalization of Hardy's uncertainty principle to vector-valued functions

Establishes a connection between kernel localization and spectral properties

Abstract

In this paper, we generalise Hardy's 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.