Article ID Journal Published Year Pages File Type
4599990 Linear Algebra and its Applications 2013 30 Pages PDF
Abstract

The strong Szegő limit theorem may be formulated in terms of finite-dimensional operators of the form(PNGPN)n−PNGnPNfor n=1,2,…, where GG denotes the operator of multiplication by a suitably restricted d×dd×d mvf (matrix-valued function) acting on the space of d×1d×1 vvfʼs (vector-valued functions) f   that meet the constraint ∫02πf(eiθ)⁎Δ(eiθ)f(eiθ)dθ<∞, where Δ(eiθ)=IdΔ(eiθ)=Id and PNPN denotes the orthogonal projection onto the space of trigonometric vector polynomials of degree at most N   that are subject to the same summability constraint. In this paper, we study these operators for a class of mvfʼs Δ which admit factorizations Δ(eiθ)=Q(eiθ)⁎Q(eiθ)=R(eiθ)R(eiθ)⁎Δ(eiθ)=Q(eiθ)⁎Q(eiθ)=R(eiθ)R(eiθ)⁎, where Q±1Q±1, R±1R±1 belong to the Wiener plus algebra of d×dd×d mvfʼs on the unit circle. We show thatκn(G)=deflimN↑∞trace{(PNGPN)n−PNGnPN} exists and is independent of Δ when the commutativity conditions GQ=QGGQ=QG and R⁎G=R⁎GR⁎G=R⁎G are in force. The space of trigonometric vector polynomials of degree at most N is identified as a de Branges reproducing kernel Hilbert space of vector polynomials of degree at most N   and weighted analogs of the strong Szegő limit theorem are established. If Q−1Q−1 and R−1R−1 are matrix polynomials, then the inverse of the block Toeplitz matrix corresponding to Δ is of the band type. Explicit formulas for trace{PNGnPN}trace{PNGnPN} are obtained in this case.

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Physical Sciences and Engineering Mathematics Algebra and Number Theory
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