## Abstract

The strong Szego limit theorem may be formulated in terms of finite-dimensional operators of the form(P_{N}GP_{N}) ^{n}-P_{N}G^{n}P_{N}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 ∫_{0}^{2π}f(e ^{iθ})*Δ(e^{iθ})f(e^{iθ}) dθ<∞, where Δ(e^{iθ})=^{Id} and P_{N} 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 mvf's Δ which admit factorizations Δ(e^{iθ})=Q(e ^{iθ})*Q(e^{iθ})=R(e^{iθ}) R(e^{iθ})*, where Q±^{1}, R±^{1} belong to the Wiener plus algebra of d×d mvf's on the unit circle. We show that^{κn}(G)= deflimN↑∞trace{(P_{N}GP_{N})n-P_{N}GnP _{N}} 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 Szego 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{P_{N}GnP_{N}} are obtained in this case.

Original language | English |
---|---|

Pages (from-to) | 3070-3099 |

Number of pages | 30 |

Journal | Linear Algebra and Its Applications |

Volume | 439 |

Issue number | 10 |

DOIs | |

State | Published - 15 Nov 2013 |

Externally published | Yes |

## Keywords

- De Branges spaces of vector polynomials
- Strong Szego limit theorem
- Trace formulas

## ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics