Related papers: Toeplitz condition numbers as an $H^{\infty}$ inte…

We study the structure of isometries defined on the algebra $\mathcal{A}$ of upper-triangular Toeplitz matrices. Our first result is that a continuous multiplicative isometry $\mathcal{A}\to M_n$ must be of the form either $A\mapsto UAU^*$…

We prove upper and lower bounds on the number of pairs of commuting $n\times n$ matrices with integer entries in $[-T,T]$, as $T\to \infty$. Our work uses Fourier analysis and leads us to an analysis of exponential sums involving matrices…

A well known result of C. Cowen states that, for a symbol $\varphi \in L^{\infty }, \; \varphi \equiv \bar{f}+g \;\;(f,g\in H^{2})$, the Toeplitz operator $T_{\varphi }$ acting on the Hardy space of the unit circle is hyponormal if and only…

A classical result by R. Rochberg says that every bounded Toeplitz operator $T$ on the Hilbert Paley-Wiener space $\mathrm{PW}_a^2$ admits a bounded symbol $\varphi$. We generalize this result to Toeplitz operators on the Banach…

Motivated by applications in single-cell biology and metagenomics, we investigate the problem of matrix reordering based on a noisy disordered monotone Toeplitz matrix model. We establish the fundamental statistical limit for this problem…

In this paper, we study the linear mapping which sends the sequence $x=(x_n)_{n \in \mathbb{N}}$ to $y=(y_n)_{n \in \mathbb{N}}$ where $y_n = \sum_{k=1}^\infty f(n/k)x_k$ for $f: \mathbb{Q}^+ \to \mathbb{C}$. This operator is the…

Asymptotic forms of the Hilbert-Scmidt and Hilbert norms of positive definite Toeplitz matrices $Q_{N}=(b(j-k))_{j,k=0}^{N-1}$ as $N\to \infty $ are determined. Here $b(j)$ are consequent trigonometric moments of a generating non-negative…

Unitary representations of the Temperley-Lieb algebra $TL_N(Q)$ on the tensor space $({\mathbb C^n})^{\otimes N}$ are considered. Two criteria are given for determining when an orthogonal projection matrix $P$ of a rank $r$ gives rise to…

The paper gives the background for Toeplitz $T_a$ and Hankel $H_a$ operators acting between distinct Hardy type spaces over the unit circle $\mathbb{T}$. We characterize possible symbols of such operators and prove general versions of…

Let $X$ be a Banach function space over the unit circle such that the Riesz projection $P$ is bounded on $X$ and let $H[X]$ be the abstract Hardy space built upon $X$. We show that the essential norm of the Toeplitz operator $T(a):H[X]\to…

It is a classical result that, if a maximal symmetric operator $T$ in a Krein space $\mathcal{H}=\mathcal{H}^-[\oplus]\mathcal{H}^+$ has the property $\mathcal{H}^-\subseteq\mathcal{D}_T$, then the imaginary part of its eigenvalue $\lambda$…

In this paper we derive a Toeplitz-structured closed form of the unique positive semi-definite stabilizing solution for the discrete-time algebraic Riccati equations, especially for the case that the state matrix is not stable. Based on the…

The classical as well as non commutative Korovkin-type theorems deal with convergence of positive linear maps with respect to modes of convergences such as norm convergence and weak operator convergence. In this article, Korovkin-type…

This survey paper deals with upper and lower bounds on the number of $k$-matchings in regular graphs on $N$ vertices. For the upper bounds we recall the upper matching conjecture which is known to hold for perfect matchings. For the lower…

We present a sublinear time algorithm for computing a near optimal low-rank approximation to any positive semidefinite (PSD) Toeplitz matrix $T\in \mathbb{R}^{d\times d}$, given noisy access to its entries. In particular, given entrywise…

This article focuses on the fluctuations of linear eigenvalue statistics of $T_{n\times p}T'_{n\times p}$, where $T_{n\times p}$ is an $n\times p$ Toeplitz matrix with real, complex or time-dependent entries. We show that as $n \rightarrow…

Histopolation is the approximation procedure that associates a degree $ d-1 $ polynomial $ p_{d-1} \in \mathscr{P}_{d-1} (I) $ with a locally integrable function $ f $ imposing that the integral (or, equivalently, the average) of $p$…

The purpose of this paper is to compute the asymptotics of determinants of finite sections of operators that are trace class perturbations of Toeplitz operators. For example, we consider the asymptotics in the case where the matrices are of…

Let $M_n$ be an $n\times n$ random matrix with i.i.d. Bernoulli(p) entries. We show that there is a universal constant $C\geq 1$ such that, whenever $p$ and $n$ satisfy $C\log n/n\leq p\leq C^{-1}$, \begin{align*} {\mathbb…

Given two univalent harmonic mappings $f_1$ and $f_2$ on $\mathbb{D}$, which lift to minimal surfaces via the Weierstrass-Enneper representation theorem, we give necessary and sufficient conditions for $f_3=(1-s)f_1+sf_2$ to lift to a…