Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6415218 | Journal of Functional Analysis | 2014 | 20 Pages |
Continuous analogs of the strong SzegÅ limit theorem may be formulated in terms of operators of the form (PTGPT)nâPTGnPT for n=1,2,â¦, where G denotes the operator of multiplication by a suitably restricted dÃd mvf (matrix-valued function) acting on the space of dÃ1 vvfʼs (vector-valued functions) f that meet the constraint â«âââf(μ)âÎ(μ)f(μ)dμ<â with Î(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 Î of the form Î(μ)=Q(μ)âQ(μ)=R(μ)R(μ)â, where Q±1, R±1 are dÃd mvfʼs in the Wiener plus algebra. This paper extends an earlier paper [6] by replacing the assumption that eiTλQ(Râ)â1 is an inner mvf for some T⩾0 by the less restrictive assumption that the Hankel operator with symbol Q(Râ)â1 is compact. We show that (PTGPT)nâPTGnPT is a trace-class operator, thatκn(G)=deflimTââtrace{(PTGPT)nâPTGnPT} exists and is independent of Q and R when GQ=QG and GRâ=RâG. An example which shows that κn(G) may depend on Q and R if these commutation conditions are not in force is furnished.