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

Continuous analogs of the strong Szegő limit theorem may be formulated in terms of operators of the form (PTGPT)n−PTGnPT for 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 vvfʼs (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 vvfʼs 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)=Q(μ)⁎Q(μ)=R(μ)R(μ)⁎, where Q±1, R±1 are d×d mvfʼs in the Wiener plus algebra. This paper extends an earlier paper [6] by replacing the assumption that eiTλQ(R⁎)−1 is an inner mvf for some T⩾0 by the less restrictive assumption that the Hankel operator with symbol Q(R⁎)−1 is compact. We show that (PTGPT)n−PTGnPT is a trace-class operator, thatκn(G)=deflimT↑∞trace{(PTGPT)n−PTGnPT} exists and is independent of Q and R when GQ=QG and GR⁎=R⁎G. An example which shows that κn(G) may depend on Q and R if these commutation conditions are not in force is furnished.