Szegö limit theorems for operators with almost periodic diagonals

Theorems of Szegő-Verblunsky type in the multivariate and almost periodic settings

The classical Szegő-Verblunsky theorem relates integrability of the logarithm of the absolutely continuous part of a probability measure on the circle to square summability of the sequence of recurrence coefficients for the orthogonal polynomials determined by the measure. The present paper constructs orthogonal polynomials on the torus of arbitrary finite dimension in order to prove theorems of Szegő-Verblunsky type in the multivariate and almost periodic settings. The results are applied to the one-dimensional Schrödinger equation in impedance form to yield a new trace formula valid for piec…

对角算子与置换算子的周期点

Periodic Points of Diagonal and Permutation Operators

本文研究希尔伯特空间上正规算子周期点的存在性条件，刻画了以全空间为周期点的正规算子，并分析了对角算子与置换算子的周期点结构，证明相关对角算子的稠密性。

Asymptotics of determinants for finite sections of operators with almost periodic diagonals

Let $A = (a_{j,k})_{j,k=-\infty}^\infty$ be a bounded linear operator on $l^2(\mathbb{Z})$ whose diagonals $D_n(A) = (a_{j,j-n})_{j=-\infty}^\infty\in l^\infty(\mathbb{Z})$ are almost periodic sequences. For certain classes of such operators and under certain conditions, we are going to determine the asymptotics of the determinants $\det A_{n_1,n_2}$ of the finite sections of the operator $A$ as their size $n_2 - n_1$ tends to infinity. Examples of such operators include block Toeplitz operators and the almost Mathieu operator.

A Connes-amenable, dual Banach algebra need not have a normal, virtual diagonal

Let $G$ be a locally compact group, and let $WAP(G)$ denote the space of weakly almost periodic functions on $G$. We show that, if $G$ is a $[SIN]$-group, but not compact, then the dual Banach algebra $WAP(G)^\ast$ does not have a normal, virtual diagonal. Consequently, whenever $G$ is an amenable, non-compact $[SIN]$-group, $WAP(G)^\ast$ is an example of a Connes-amenable, dual Banach algebra without a normal,virtual diagonal.

Sharp spectral estimates for periodic matrix-valued Jacobi operators

For the periodic matrix-valued Jacobi operator $J$ we obtain the estimate of the Lebesgue measure of the spectrum $|\s(J)|\le4 \min_n\Tr(a_na_n^*)^\frac12$, where $a_n$ are off-diagonal elements of $J$. Moreover estimates of width of spectral bands are obtained.