Szegö Limit Theorem on the Lattice

In this paper, we prove a Szegö type limit theorem on l(Z). We consider operators of the form H = ρ∆+|ξ|, 0 ≤ ρ, 0 < k < 2 on l(Z) and πλ the orthogonal projection of l (Z) on to the space of eigenfunctions of H with eigenvalues ≤ λ. We take A be a 0th order self adjoint pseudo difference operator with symbol a(ξ, x) satisfying [A,H ](H +1) bounded for some 0 < σ < 1 2 . Then for f ∈ C(R) and (ξ, x) ∈ Z d × T, lim λ→∞ tr f(πλAπλ) rank πλ = lim λ→∞ 1 (2π)d 1 vol(h(ξ, x) ≤ λ) ∑