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Let α, β ∈ (0,1) and Ka,b : = { a Î L¥ (\mathbbT):åk = 1¥ | [^(a)]( - k) |2 k2a ¥ ,åk = 1¥ | [^(a)](k) |2 k2b ¥ }.K^{\alpha ,\beta } : = \left\{ {a \in L^\infty (\mathbb{T}):\sum\limits_{k = 1}^\infty {\left| {\hat a( - k)} \right|^2 k^{2\alpha } < \infty } ,\sum\limits_{k = 1}^\infty {\left| {\hat a(k)} \right|^2 k^{2\beta } < \infty } } \right\}. Mark Krein proved in 1966 that K1/2,1/2 forms a Banach algebra. He also observed that this algebra is important in the asymptotic theory of finite Toeplitz matrices. Ten years later, Harold Widom extended earlier results of Gabor Szegő for scalar symbols and established the asymptotic trace formula trace f(Tn (a)) = (n + 1)Gf (a) + Ef (a) + o(1) as n ® ¥trace f(T_n (a)) = (n + 1)G_f (a) + E_f (a) + o(1) as n \to \infty for finite Toeplitz matrices Tn(a) with matrix symbols a ∈ K N×N 1/2,1/2. We show that if α + β ≥ 1 and a ∈ K N×N α,β then the Szegő-Widom asymptotic trace formula holds with o(1) replaced by o(n1-α-β).

Asymptotics of Toeplitz Matrices with Symbols in Some Generalized Krein Algebras