Trace formulas for a class of vector-valued Wiener-Hopf like operators, I

Continuous analogs of the strong Szego limit theorem may be formulated in terms of operators of the form (P_TGP_T)ⁿ-P_TGⁿP_T, forn=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 ∫f(μ)^*δ (μ) f (μ) dμ < ∞ with δ (μ) = _{I d} and P_T denotes the orthogonal projection onto the space of entire vvf's of exponential type ≤T that are subject to the same summability constraint. In this paper we study these operators for a more general class of δ of the form, in which h is a d × d summable mvf and δ is positive definite for every μ∈R. We show that (P_TGP_T)n-P_TGnP_T is trace-class, when T is sufficiently large, and lim_T↑∞trace{(P_TGP_T)n-P_TGnP_T} exists and is independent of h when G commutes with certain factors of δ. This extends the results of the first author who considered analogous problems with δ (μ) = δ (μ) I_d, a scalar multiple of I_d.