Trace formulas for a class of truncated block Toeplitz operators Dedicated to Harm Bart on the occasion of his 70th birthday

The strong Szego limit theorem may be formulated in terms of finite-dimensional operators of the form(P_NGP_N) ⁿ-P_NGⁿP_Nfor n=1,2,., where G denotes the operator of multiplication by a suitably restricted d×d mvf (matrix-valued function) acting on the space of d×1 vvf's (vector-valued functions) f that meet the constraint ∫₀^2πf(e ^iθ)*Δ(e^iθ)f(e^iθ) dθ<∞, where Δ(e^iθ)=^Id and P_N denotes the orthogonal projection onto the space of trigonometric vector polynomials of degree at most N that are subject to the same summability constraint. In this paper, we study these operators for a class of mvf's Δ which admit factorizations Δ(e^iθ)=Q(e ^iθ)*Q(e^iθ)=R(e^iθ) R(e^iθ)*, where Q±¹, R±¹ belong to the Wiener plus algebra of d×d mvf's on the unit circle. We show that^κn(G)= deflimN↑∞trace{(P_NGP_N)n-P_NGnP _N} exists and is independent of Δ when the commutativity conditions GQ=QG and R*G=R*G are in force. The space of trigonometric vector polynomials of degree at most N is identified as a de Branges reproducing kernel Hilbert space of vector polynomials of degree at most N and weighted analogs of the strong Szego limit theorem are established. If Q-¹ and R-¹ are matrix polynomials, then the inverse of the block Toeplitz matrix corresponding to Δ is of the band type. Explicit formulas for trace{P_NGnP_N} are obtained in this case.