Asymptotics of Toeplitz Matrices with Symbols in Some Generalized Krein Algebras

Let alpha,beta is an element of (0,1) and K-alpha,K-beta := {a is an element of L-infinity(T): Sigma(infinity)(k=1) vertical bar(a) over cap(-k)vertical bar(2)k(2 alpha) < infinity, Sigma(infinity)(k=1) vertical bar(a) over cap (k)vertical bar(2)k(2 beta) < infinity}. Mark Krein proved in 1966 that K-1/2,K-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 Szego for scalar symbols and established the asymptotic trace formula trace f(T-n(a)) = (n+1)G(f)(a) + E-f(a) + o(1) as n ->infinity for finite Toeplitz matrices T-n(a) with matrix symbols a is an element of K-NXN(1/2,1/2). We show that if alpha + beta >= 1 and a is an element of K-NXN(alpha,beta), then the Szego-Widom asymptotic trace formula holds with o(1) replaced by o(n(1-alpha-beta)).