Trace formulas for a class of truncated block Toeplitz operators

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 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 ∫02πf(eiθ)⁎Δ(eiθ)f(eiθ)dθ<∞, where Δ(eiθ)=Id and PN 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)⁎, where Q±1, R±1 belong to the Wiener plus algebra of d×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=QG and R⁎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−1 and R−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} are obtained in this case.