A Trace Formula for a Class of Variable-Coefficient Block Toeplitz Matrices

This paper is concerned with spectral asymptotics for variable coeefficient block Toeplitz matrices op(n) (sigma) given by 1/2(pi) (integral) (sigma)(j/n, (theta) e^(-i)(j-k)(theta} d(theta) (j,k=0,1,…, n), where (sigma)(x, e^{i(theta)) is a matrix-valued function of fixed order defined on [0,1] * T . More precisely, we compute the second-order asymptotics of the trace of f((op)(n),(sigma)) , where f belongs to a suitable class of functions; (tr) f((op)n(sigma)=c1n + c2 (log) n + o((log) n) as n – (infinity) , where c1,c2 are constants given by explicit formulas.