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

Continuous analogs of the strong Szegő limit theorem may be formulated in terms of operators of the form (PTGPT)n−PTGnPT,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)=Id and PT 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 Δ(μ)=Id+∫−∞∞eiμxh(x)dx, in which h is a d×d summable mvf and Δ is positive definite for every μ∈R. We show that (PTGPT)n−PTGnPT is trace-class, when T is sufficiently large, and limT↑∞trace{(PTGPT)n−PTGnPT} 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)Id, a scalar multiple of Id.