Uncertainty principles for compact groups

Abstract

We establish an uncertainty principle over arbitrary compact groups, generalizing several previous results. Specifically, we show that if P and R are operators on L2(G) such that P commutes with projection onto every measurable subset of G and R commutes with left-multiplication by elements of G, then ||PR|| ≤ ||P · χG||2||R||2, where χG: g → 1 is the characteristic function of G. As a consequence, we show that every nonzero function f in L2(G) satisfies μ(suppf) · ∑ρ∈Ĝ dρ rank f(ρ) ≥ 1. © 2009 University of Illinois.