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

Continuous analogs of the strong Szego 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 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 δ(μ)=Q(μ)^*Q(μ)=R(μ)R(μ)^*, where Q^±1, R^±1 are d×d mvf's 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 κ_n(G) may depend on Q and R if these commutation conditions are not in force is furnished.