## Illinois Journal of Mathematics

### 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 $\mu(\operatorname{\mathbf{supp}} f)\cdot \sum_{\rho\in\hat G} d_\rho\operatorname{\mathbf{rank}} \hat f(\rho) \geq1$.

#### Article information

Source
Illinois J. Math., Volume 52, Number 4 (2008), 1315-1324.

Dates
First available in Project Euclid: 18 November 2009

https://projecteuclid.org/euclid.ijm/1258554365

Digital Object Identifier
doi:10.1215/ijm/1258554365

Mathematical Reviews number (MathSciNet)
MR2595770

Zentralblatt MATH identifier
1184.43004

#### Citation

Alagic, Gorjan; Russell, Alexander. Uncertainty principles for compact groups. Illinois J. Math. 52 (2008), no. 4, 1315--1324. doi:10.1215/ijm/1258554365. https://projecteuclid.org/euclid.ijm/1258554365

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