Toeplitz Operators on Locally Compact Abelian Groups

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Mathematics (HMC)

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Let G be a non-discrete locally compact Abelian group, let Γ be its non-compact dual group, and let m and μ be Haar measures on G and Γ respectively, normalized so the Fourier inversion theorem holds. 1f fɛL1(G), F=f is the Fourier transform of f, and W is a Borel set of Γ, with compact closure, then the finite Toeplitz operator FW on L2(W) generated by f is defined by (FWϕ)(γ)=∫W F(γ−τ)ϕ(τ)dτ. In the case in which Γ is compactly generated, there exist sequences {Wn} of Borel sets of Γ, with compact closure, such that if fɛL1(G) is real valued, F Wn:L2(Wn)→L2(Wn) is the completely continuous self-adjoint finite Toeplitz operator generated by f and λ nj, j=1,2,3,⋯, are the corresponding eigenvalues, then if 0ɛ[a,b] and


we have

lim_n→∞ # of λ njɛ[a,b] / μ(Wn)=m{x:a≦f(x)≦b}.

This asymptotic eigenvalue distribution theorem is extended in several ways. First of all, the class of sequences {Wn} for which the result holds is examined and found to be quite extensive. Secondly, several generalizations of the finite Toeplitz operator generated by a function fɛL1(G) are considered and corresponding results on the asymptotic distribution of eigenvalues are obtained for most of these operators. Finally, a further extension of these results is made to the case in which Γ is not compactly generated.

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© 1965 Indiana University Mathematics Journal