We study the spectrum of the product of two Toeplitz operators. Assume that the symbols of these operators are continuous and real-valued and that one of them is non-negative. We prove that the spectrum of the product of finite section Toeplitz matrices converges to the spectrum of the product of the semi-infinite Toeplitz operators. We give an example showing that the supremum of this set is not always the supremum of the product of the two symbols. Finally, we provide an application in probability which is the first motivation of this study. More precisely, we obtain a large deviation principle for Gaussian quadratic forms.
We introduce and give a more or less complete study of a family of branching-Toeplitz operators on the Hilbert space $\ell^2(T_q)$ indexed by a rooted homogeneous tree $T_q$ of degree $q\ge 2$. The finite dimensional analogues of such operators form a very natural family of structured sparse matrices called branching-Toeplitz matrices and will also be investigated. The branching-Toeplitz operators/matrices in this paper should be viewed as natural generalizations of the standard Toeplitz operators/matrices. We will apply our results to construct a family of determinantal point processes on hom…
The maximal commutative subalgebras containing only Toeplitz matrices have been identified as generalized circulants. A similar simple description cannot be obtained for block Toeplitz matrices. We introduce and investigate certain families of maximal commutative algebras of block Toeplitz matrices.
Toeplitz matrices are ubiquitous and play important roles across many areas of mathematics. In this paper, we present some algebraic results concerning block Toeplitz matrices with block entries belonging to a commutative algebra $Å$. The characterization of normal block Toeplitz matrices with entries from $Å$ is also obtained.
Chien, Liu, Nakazato, and Tam proved that all n by n classical Toeplitz matrices (one-level Toeplitz matrices) are unitarily similar to complex symmetric matrices via two types of unitary matrices and the type of the unitary matrices only depends on the parity of n. In this paper, we extend their result to multilevel Toeplitz matrices that any multilevel Toeplitz matrix is unitarily similar to a complex symmetric matrix. We provide a method to construct the unitary matrices that uniformly turn any multilevel Toeplitz matrix to a complex symmetric matrix by taking tensor products of these two t…
The maximal algebras of scalar Toeplitz matrices are known to be formed by generalized circulants. The identification of algebras consisting of block Toeplitz matrices is a harder problem, that has received little attention up to now. We consider the case when the block entries of the matrices belong to a commutative algebra $ \mathcal{A} $. After obtaining some general results, we classify all the maximal algebras for certain particular cases of $ \mathcal{A}$.
Conditions are established under which Fredholmness, Coburn's property and one- or two-sided invertibility are shared by a Toeplitz operator with matrix symbol $G$ and the Toeplitz operator with scalar symbol $\det G$. These results are based on one-sided invertibility criteria for rectangular matrices over appropriate commutative rings and related scalar corona type problems.
The results on the inversion of convolution operators and Toeplitz matrices in the 1-D (one dimensional) case are classical and have numerous applications. We consider a 2-D case of Toeplitz-block Toeplitz matrices, describe a minimal information, which is necessary to recover the inverse matrices, and give a complete characterisation of the inverse matrices.
This work provides two results obtained as a consequence of an inversion formula for Toeplitz matrices with real symbol. First we obtain an asymptotic expression for the minimal eigenvalues of a Toeplitz matrix with a symbolwhich is periodic, even and derivable on $[0, 2π[$. Next we prove that a Toeplitz band matrix with a symbol without zeros on the united circle is invertible with an inverse which is essentially a band matrix. As a consequence of this last statement we give an asymptotic estimation for the entries of the inverse of a Toeplitz matrix with a regular symbol.
In this paper we establish several relations between the determinants of the following structured matrices: Hankel matrices, symmetric Toeplitz + Hankel matrices and Toeplitz matrices. Using known results for the asymptotic behavior of Toeplitz determinants, these identities are used in order to obtain Fisher-Hartwig type results on the asymptotics of certain skewsymmetric Toeplitz determinants and certain Hankel determinants.
We establish asymptotic formulas for the determinants of finite Toeplitz + Hankel matrices of size N, as N goes to infinity for singular generating functions defined on the unit circle in the special case where the generating function is even, i.e., where the Toeplitz + Hankel matrices are symmetric.
Based on a question raised by N. Feldman we discuss the dynamics of tuples of upper triangular Toeplitz matrices over $\mathbb{C}$. Some open problems are posed.
We consider boundary conditions of self-adjoint banded Toeplitz matrices. We ask if boundary conditions exist for banded self-adjoint Toeplitz matrices which satisfy operator inequalities of Dirichlet-Neumann bracketing type. For a special class of banded Toeplitz matrices including integer powers of the discrete Laplacian we find such boundary conditions. Moreover, for this class we give a lower bound on the spectral gap above the lowest eigenvalue.
We prove a higher order asymptotic formula for traces of finite block Toeplitz matrices with symbols belonging to Hölder-Zygmund spaces. The remainder in this formula goes to zero very rapidly for very smooth symbols. This formula refine previous asymptotic trace formulas by Szegö and Widom and complement higher order asymptotic formulas for determinants of finite block Toeplitz matrices due to Böttcher and Silbermann.
A recent paper of A.~Connes and W.D.~van Suijlekom identifies the operator system of $n\times n$ Toeplitz matrices with the dual of the space of all trigonometric polynomials of degree less than $n$. The present paper examines this identification in somewhat more detail by showing explicitly that the Connes--van Suijlekom isomorphism is a unital complete order isomorphism of operator systems. Consequences of this complete order isomorphism are also examined, yielding two special results of note: (i) that every positive linear map of the $n\times n$ complex matrices is completely positive when …
The purpose of this paper is to describe asymptotic formulas for determinants of a sum of finite Toeplitz and Hankel matrices with singular generating functions. The formulas are similar to those of the analogous problem for finite Toeplitz matrices for a certain class of symbols. However, the appearance of the Hankel matrices changes the nature of the asymptotics in some instances depending on the location of the singularities. Several concrete examples are also described in the paper.
We show that an infinite Toeplitz+Hankel matrix $T(φ) + H(ψ)$ generates a bounded (compact) operator on $\ell^p(\mathbb{N}_0)$ with $1\leq p\leq \infty$ if and only if both $T(φ)$ and $H(ψ)$ are bounded (compact). We also give analogous characterizations for Toeplitz+Hankel operators acting on the reflexive Hardy spaces. In both cases, we provide an intrinsic characterization of bounded operators of Toeplitz+Hankel form similar to the Brown-Halmos theorem. In addition, we establish estimates for the norm and the essential norm of such operators.
We derive novel explicit formulas for the inverses of truncated block Toeplitz matrices that correspond to a multivariate minimal stationary process. The main ingredients of the formulas are the Fourier coefficients of the phase function attached to the spectral density of the process. The derivation of the formulas is based on a recently developed finite prediction theory applied to the dual process of the stationary process. We illustrate the usefulness of the formulas by two applications. The first one is a strong convergence result for solutions of general block Toeplitz systems for a mult…
We compute the asymptotics of the determinants of certain $n\times n$ Toeplitz + Hankel matrices $T_n(a)+H_n(b)$ as $n\to\infty$ with symbols of Fisher-Hartwig type. More specifically we consider the case where $a$ has zeros and poles and where $b$ is related to $a$ in specific ways. Previous results of Deift, Its and Krasovsky dealt with the case where $a$ is even. We are generalizing this in a mild way to certain non-even symbols.
We review some history and some recent results concerning Toeplitz determinants and their applications. We discuss, in particular, the crucial role of the two-dimensional Ising model in stimulating the development of the theory of Toeplitz determinants.
We describe the asymptotics of the spectral norm of finite Toeplitz matrices generated by functions with Fisher-Hartwig singularities as the matrix dimension goes to infinity. In the case of positive generating functions, our result provides the asymptotics of the largest eigenvalue, which is of interest in time series with long-range memory.
We prove higher order asymptotic formulas for determinants and traces of finite block Toeplitz matrices generated by matrix functions belonging to generalized Hölder spaces with characteristic functions from the Bari-Stechkin class. We follow the approach of Böttcher and Silbermann and generalize their results for symbols in standard Hölder spaces.
It was shown in a series of recent publications that the eigenvalues of $n\times n$ Toeplitz matrices generated by so-called simple-loop symbols admit certain regular asymptotic expansions into negative powers of $n+1$. On the other hand, recently two of the authors considered the pentadiagonal Toeplitz matrices generated by the symbol $g(x)=(2\sin(x/2))^4$, which does not satisfy the simple-loop conditions, and derived asymptotic expansions of a more complicated form. We here use these results to show that the eigenvalues of the pentadiagonal Toeplitz matrices do not admit the expected regula…
We establish connections between Schur parameters of the Schur class operator-valued functions, the corresponding simple conservative realizations, lower triangular Toeplitz matrices, and Kre\uın shorted operators. By means of Schur parameters or shorted operators for defect operators of Toeplitz matrices necessary and sufficient conditions for a simple conservative discrete-time system to be controllable/observable and for a completely non-unitary contraction to be completely non-isometric/completely non-co-isometric are obtained. For the Schur problem a characterization of central solution a…
The authors analyze the asymptotics of eigenvalues of Toeplitz matrices with certain continuous and discontinuous symbols. In particular, the authors prove a conjecture of Levitin and Shargorodsky on the near-periodicity of Toeplitz eigenvalues.
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 the form $ (a_{i-j} \pm a_{i+j+1-k})_{i,j=0... N-1} $ with $k$ is fixed. We will show that this example as well as some general classes of operators have expansions that are similar to those that appear in the Strong Szegö Limit Theorem. We also obtain exact identitities for some of the determinants that are analogous to the one derived independently by Ger…
Let $f$ be a regular real-valued non-constant symbol defined on the one dimensional torus ${\mathbb T}$. Denote respectively by $κ$ and $T$, its set of critical points and the associated Toeplitz matrix on $l^2({\mathbb N})$. If $V$ is a suitable compact perturbation, we prove that the operator $T+V$ has no singular continuous spectrum and only finite point spectrum away from the set of thresholds $f(κ)$. We also obtain some propagation estimates and apply these results to concrete examples.
The paper is devoted to exact and asymptotic formulas for the determinants of Toeplitz matrices with perturbations by blocks of fixed size in the four corners. If the norms of the inverses of the unperturbed matrices remain bounded as the matrix dimension goes to infinity, then standard perturbation theory yields asymptotic expressions for the perturbed determinants. This premise is not satisfied for matrices generated by so-called Fisher-Hartwig symbols. In that case we establish formulas for pure single Fisher-Hartwig singularities and for Hermitian matrices induced by general Fisher-Hartwig…
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…
We consider the class of positive bounded and semi-continuous functions defined on the two dimensional torus If f belongs to this class, then f will be considered as the symbol of a Toeplitz operator truncated on a triangle parametrised by an integer number . We develop a geometric structure of the inverse of the Toeplitz operator and give an asymptotical development of the trace of its inverse wich brings out the geometry of the triangle. The foundation of this result consists in the possibility of f having a factorisation of type |g|^2 where the spectrum of g will be localised in a given sem…
For each non-negative integer $n$ let $\mathcal{A}_n$ be an $n+1$ by $n+1$ Toeplitz matrix over a finite field, $F$, and suppose for each $n$ that $\mathcal{A}_n$ is embedded in the upper left corner of $\mathcal{A}_{n+1}$. We study the structure of the sequence $ν= \{ν_n :n \in \mathbb{Z}^+\}$, where $ν_n = \text{null}(\mathcal{A}_n)$ is the nullity of $\mathcal{A}_{n}$. For each $n\in \mathbb{Z}^+$ and each nullity pattern $ν_0,ν_1,\dots,ν_n$, we count the number of strings of Toeplitz matrices $\mathcal{A}_0,\mathcal{A}_1,\dots,\mathcal{A}_{n}$ with this pattern. As an application we presen…
Motivated by [9] we study the existence of the inverse of infinite Hermitian moment matrices associated with measures with support on the complex plane. We relate this problem to the asymptotic behaviour of the smallest eigenvalues of finite sections and we study it from the point of view of infinite transition matrices associated to the orthogonal polynomials. For Toeplitz matrices we introduce the notion of weakly asymptotic Toeplitz matrix and we show that, under certain assumptions, the inverse of a Toeplitz moment matrix is weakly asymptotic Toeplitz. Such inverses are computed in terms o…
In this paper, we will consider a noncommutative probability space, $(T,E),$ over a Toeplitz matricial algebra $B=\QTR{cal}{C}^{N},$ for $N\in \QTR{Bbb}{N}$, induced by a (scalar-valued) noncommutative probability space, $(A,\QTR{cal}ϕ),$ with a suitable $B$-functional, $E:T\to B,$ defined by $ϕ.$ On this framework, we will observe amalgamated R-transform calculus with respect to a $B$-functional, $E.$ The technique and idea are came from those in [12] and Free Probability of type $B$ studied in [31].
The condition numbers $CN(T)==||T|| .||T^{-1}|| $ of Toeplitz and analytic $n\times n$ matrices $T$ are studied. It is shown that the supremum of $CN(T)$ over all such matrices with $||T|| \leq1$ and the given minimum of eigenvalues $r=min_{i=1..n}|λ_{i}|>0$ behaves as the corresponding supremum over all $n\times n$ matrices (i.e., as $\frac{1}{r{}^{n}}$ (Kronecker)), and this equivalence is uniform in $n$ and $r$. The proof is based on a use of the Sarason-Sz.Nagy-Foias commutant lifting theorem.
Toeplitz matrices for the study of the fractional Laplacian on a bounded interval. In this work we get a deep link between (--$Δ$) $α$ ]0,1[ the fractional Laplacian on the interval ]0, 1[ and T N ($Φ$ $α$) the Toeplitz matrices of symbol $Φ$ $α$ : $θ$ $\rightarrow$ |1 -- e i$θ$ | 2$α$ when N goes to the infinity and for $α$ $\in$]0, 1 2 [$\cup$] 1 2 , 1[. In the second part of the paper we provide a Green function for the fractional equation (--$Δ$) $α$ ]0,1[ ($ψ$) = f for $α$ $\in$]0, 1 2 [ and f a sufficiently smooth function on [0, 1]. The interest is that this Green's function is the same…
Let $α,β\in(0,1)$ and \[ K^{α,β}:=\left\{a\in L^\infty(\T): \sum_{k=1}^\infty |\hat{a}(-k)|^2 k^{2α}<\infty, \sum_{k=1}^\infty |\hat{a}(k)|^2 k^{2β}<\infty \right\}. \] Mark Krein proved in 1966 that $K^{1/2,1/2}$ forms a Banach algebra. He also observed that this algebra is important in the asymptotic theory of finite Toeplitz matrices. Ten years later, Harold Widom extended earlier results of Gabor Szegő for scalar symbols and established the asymptotic trace formula \[ \operatorname{trace}f(T_n(a))=(n+1)G_f(a)+E_f(a)+o(1) \quad\text{as}\ n\to\infty \] for finite Toeplitz matrices $T_n(a)$ w…
In [V. M. Abramov, \emph{Bull. Aust. Math. Soc.} \textbf{104} (2021), 108--117] the fixed point equation for an infinite nonnegative Toeplitz matrix has been studied. It was found the conditions for existence of a positive solution and bounded positive solution. However, the proof of the existence of a positive solution was entirely straightforward, not admitting extensions for more general types of matrices. In the present note, we provide an alternative proof for the existence of a positive solution in more general case. The presented proof is based on an application of a variant of the fixe…
We express minors of Toeplitz matrices of finite and large dimension in terms of symmetric functions. Comparing the resulting expressions with the inverses of some Toeplitz matrices, we obtain explicit formulas for a Selberg-Morris integral and for specializations of certain skew Schur polynomials.
