Uncertainty principles for compact groups

Abstract

We establish an uncertainty principle over arbitrary compact groups, generalizing several previous results. Specifically, we show that if $\mathrm{P}$ and $\mathrm{R}$ are operators on $L^2(G)$ such that $\mathrm{P}$ commutes with projection onto every measurable subset of $G$ and $\mathrm{R}$ commutes with left-multiplication by elements of $G$, then $\|\operatorname{PR}\| \leq\|\mathrm{P} \cdot\chi_G \|_2 \|\mathrm {R}\|_2$, where $\chi_G : g \mapsto1$ is the characteristic function of $G$. As a consequence, we show that every nonzero function $f$ in $L^2(G)$ satisfies $\mu(\operatorname{\mathbf{supp}} f)\cdot \sum_{\rho\in\hat G} d_\rho\operatorname{\mathbf{rank}} \hat f(\rho) \geq1$.