Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4672883 | Indagationes Mathematicae | 2012 | 25 Pages |
Abstract
Continuous analogs of the strong SzegÅ limit theorem may be formulated in terms of operators of the form (PTGPT)nâPTGnPT,forn=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 vvf's (vector-valued functions) f that meet the constraint â«f(μ)âÎ(μ)f(μ)dμ<â with Î(μ)=Id and PT denotes the orthogonal projection onto the space of entire vvf's of exponential type â¤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)=Id+â«âââeiμxh(x)dx, in which h is a dÃd summable mvf and Î is positive definite for every μâR. We show that (PTGPT)nâPTGnPT is trace-class, when T is sufficiently large, and limTââtrace{(PTGPT)nâPTGnPT} exists and is independent of h when G commutes with certain factors of Î. This extends the results of the first author who considered analogous problems with Î(μ)=δ(μ)Id, a scalar multiple of Id.
Related Topics
Physical Sciences and Engineering
Mathematics
Mathematics (General)
Authors
Harry Dym, David P. Kimsey,