At present there exist numerous different approaches to results on Toeplitz determinants of the type of Szegö's strong limit theorem. The intention of this paper is to show that Jacobi's theorem on the minors of the inverse matrix remains one of the most comfortable tools for tackling the matter. We repeat a known proof of the Borodin-Okounkov formula and thus of the strong Szegö limit theorem that is based on Jacobi's theorem. We then use Jacobi's theorem to derive exact and asymptotic formulas for Toeplitz determinants generated by functions with nonzero winding number. This derivation is new and completely elementary
Truncated Toeplitz operators are compressions of multiplication operators on $L^2$ to model spaces (that is, subspaces of $H^2$ which are invariant with respect to the backward shift). For this class of operators we prove certain Szegö type theorems concerning the asymptotics of their compressions to an increasing chain of finite dimensional model spaces.
We discuss an analogue of the First Szegő Limit Theorem for multiplicative Toeplitz operators and highlight the role of the multliplicative Følner condition in this topic.
We establish versions of Szegő's distance formula and Widom's theorem on invertibility of (a family of) Toeplitz operators in a class of finite codimension subalgebras of uniform algebras, obtained by imposing a finite number of linear constraints. Each such algebra is naturally represented on a family of reproducing kernel Hilbert spaces, which play a central role in the proofs.
We study the properties of stationary G-chains in terms of their generating functions. In particular, we prove an analogue of the Szegő limit theorem for symplectic eigenvalues, derive an expression for the entropy rate of stationary quantum Gaussian processes, and study the distribution of symplectic eigenvalues of truncated block Toeplitz matrices. We also introduce a concept of symplectic numerical range, analogous to that of numerical range, and study some of its basic properties, mainly in the context of block Toeplitz operators.
We study semi-infinite Jacobi matrices $H=H_{0}+V$ corresponding to trace class perturbations $V$ of the "free" discrete Schrödinger operator $H_{0}$. Our goal is to construct various spectral quantities of the operator $H$, such as the weight function, eigenfunctions of its continuous spectrum, the wave operators for the pair $H_{0}$, $H$, the scattering matrix, the spectral shift function, etc. This allows us to find the asymptotic behavior of the orthonormal polynomials $P_{n}(z)$ associated to the Jacobi matrix $H $ as $n\to\infty$. In particular, we consider the case of $z$ inside the spe…
The Jacobi-Trudi formulas imply that the minors of the banded Toeplitz matrices can be written as certain skew Schur polynomials. In 2012, Alexandersson expressed the corresponding skew partitions in terms of the indices of the struck-out rows and columns. In the present paper, we develop the same idea and obtain some new applications. First, we prove a slight generalization and modification of Alexandersson's formula. Then, we deduce corollaries about the cofactors and eigenvectors of banded Toeplitz matrices, and give new simple proofs to the corresponding formulas published by Trench in 198…
On an identity due to ((Bump and Diaconis) and (Tracy and Widom))
对于给定符号的Toeplitz矩阵，一个经典的问题是计算其降阶到有限秩的行列式的渐近性。我们还可以考虑当移动一组初始行和列时，这些渐近性是如何受到影响的（或者，等价地，它们的子项的渐近性）。Bump and Diaconis（Toeplitz minors，J. Combin. Theory Ser. A，97（2002），pp. 252- 271）得到了涉及对称群上拉盖尔多项式和的这种移位的一个公式。他们还展示了海涅身份如何扩展到这些未成年人，这使得这个问题与随机矩阵理论有关。独立地，Tracy和Widom（On the limit of some Toeplitz-like determinants，SIAM J. Matrix Anal.应用程序、23（2002），pp. 1194- 1196）使用Wiener-Hopf分解来表示这些移位的无限矩阵的乘积。我们直接显示为什么这两个表达式是相等的，并揭示了一些结构，在这两个公式是未知的作者。我们引入一个神秘的对称函数上的微分算子，它与顶点算子非常相似。我们证明了Bump-Diaconis-Tracy-Widom恒等式是经典Jacobi-Trudi恒等式的一个微分形式。
