Abstract

Consider a discrete group G and a bounded self-adjoint convolution operator T on l²; let σ(T) be the spectrum of T. The spectral theorem gives a unitary isomorphism U between l²(G) and a direct sum ⊕_n L²(Δ_n, ν), where Δ_n ⊂ σT, and ν is a regular Borel measure supported on σ(T). Through this isomorphism T. corresponds to multiplication by the identity function on each summand. We prove that a nonzero function ƒ ∈ l²(G) and its transform U ƒ cannot be simultaneously concentrated on sets V ⊂ G, W ⊂ σ(T) such that ν(W) and the cardinality of V are both small. This can be regarded as an extension to this context of Heisenberg's classical uncertainty principle.

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