Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics

Frontmatter

Albrecht Böttcher – 20 Years of Friendship and Joint Work

Abstract

More than twenty years of our friendship with Albrecht Böttcher, as well as joint work on numerous projects (nearly 50 papers and 2 monographs!) have already passed. In the beginning of 1990s, I was already an established scientist, however his influence on me as a mathematician was very substantial.

Sergei Grudsky

Salutatory with Regards from the Mathematics Students of Chemnitz

Abstract

Dear Professor Böttcher, dear math enthusiasts.

It is a great pleasure for me – and an honor at the same time – that I’m able to express some words of gratitude and acknowledgment from the students’ point of view.

Jonas Jahns

Essay on Albrecht Böttcher

Abstract

About 40 years ago Albrecht Böttcher stepped into my life as a real person. It was when he began his study in mathematics at the TU Karl-Marx-Stadt (now TU Chemnitz) in the year 1975. My curricular duty was to give the analysis course, and so I met him for the first time at the lectures. But, as a matter of fact, I had already heard about him and his close friend Elias Wegert before.

Bernd Silbermann

Meeting Albrecht the Strong

Abstract

It happened on the 7th February 1989 in Karl-Marx-Stadt (KMS) when we met for the first time, a day which I remember with emotion. Somehow Albrecht reminded me on August den Starken (Augustus II the Strong, Elector of Saxony and King of Poland, 1670–1733) because of his strong personality and his impressive mathematical output (he had 30 publications aged 34 and was about finishing the book on Toeplitz operators with Bernd Silbermann [3]).

Frank-Olme Speck

The Beginning (the Way I Remember it)

Abstract

One Spring day of 1984, Mark Grigor’evich Krein showed me the abstract of a PhD thesis he received in the mail from Rostov-on-Don University. He liked it at first glance, so we set to read it together more carefully, and towards the end of this reading he decided to write a formal (enthusiastically positive) report to the PhD defence committee.

Ilya M. Spitkovsky

Personal Address on the Occasion of Albrecht Böttcher’s 60th Birthday

Abstract

Dear Albrecht. All the people holding this volume in their hands do this with the best wishes to your big birthday. And with this event in mind, I want to say some words.

David Wenzel

Asymptotics of Eigenvalues for Pentadiagonal Symmetric Toeplitz Matrices

Abstract

In this paper, we find uniform asymptotic formulas for all the eigenvalues of certain pentadiagonal symmetric Toeplitz matrices of large dimension. The entries of the matrices are real and we consider the case where the real-valued generating function has a minimum and a maximum such that its fourth derivative at the minimum and its second derivative at the maximum are nonzero. This is not the simple-loop case considered in [1] and [2]. We apply the main result of [7] and obtain nonlinear equations for the eigenvalues. It should be noted that our equations have a more complicated structure than the equations in [1] and [2]. Therefore, we required a more delicate method for its asymptotic analysis.

M. Barrera, S. M. Grudsky

Echelon Type Canonical Forms in Upper Triangular Matrix Algebras

Abstract

It is common knowledge that matrices can be brought in echelon form by Gaussian elimination and that the reduced echelon form of a matrix is canonical (also) in the sense that it is unique. A crucial auxiliary result in [BW] suggests a generalization of the standard echelon form. For square matrices, some new canonical forms of echelon type are introduced. One of them (suggested by observations made in [Lay] and [SW]) has the important property of being an upper triangular idempotent. The others come up when working exclusively in the context of \( \mathbb{C}^{n \times n}_{upper} \), the algebra of upper triangular n × n matrices. Subalgebras of \( \mathbb{C}^{n \times n}_{upper} \) determined by a pattern of zeros are considered too. The issue there is whether or not the canonical forms referred to above belong to the subalgebras in question. In general they do not, but affirmative answers are obtained under certain conditions on the given preorder which allow for a large class of examples and that also came up in [BES4]. Similar results hold for canonical generalized diagonal forms involving matrices for which all columns and rows contain at most one nonzero entry. The new canonical forms are used to study left, right and left/right equivalence in zero pattern algebras. For the archetypical full upper triangular case a connection with the Stirling numbers (of the second kind) and with the Bell numbers is made.

Harm Bart, Torsten Ehrhardt, Bernd Silbermann

Asymptotic Formulas for Determinants of a Special Class of Toeplitz + Hankel Matrices

Abstract

We compute the asymptotics of the determinants of certain n × n Toeplitz + Hankel matrices \( T_{n}(a)+Hn(b) \, {\rm as} \, n\rightarrow \infty \) 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.

Estelle Basor, Torsten Ehrhardt

Generalization of the Brauer Theorem to Matrix Polynomials and Matrix Laurent Series

Abstract

Given a square matrix A, Brauer’s theorem [Duke Math. J. 19 (1952), 75–91] shows how to modify one single eigenvalue of A via a rank-one perturbation, without changing any of the remaining eigenvalues. We reformulate Brauer’s theorem in functional form and provide extensions to matrix polynomials and to matrix Laurent series A(z) together with generalizations to shifting a set of eigenvalues. We provide conditions under which the modified function \( \tilde{A}(z) \) has a canonical factorization \( \tilde{A}(z)=\tilde{U}(z)\tilde{L}(z^{-1}) \) and we provide explicit expressions of the factors \( \tilde{U}(z) \) and \( \tilde{L}(z) \). Similar conditions and expressions are given for the factorization of \( \tilde{A}(z^{-1}) \). Some applications are discussed.

D. A. Bini, B. Meini

Eigenvalues of Hermitian Toeplitz Matrices Generated by Simple-loop Symbols with Relaxed Smoothness

Abstract

In a sequence of previous works with Albrecht Böttcher, we established higher-order uniform individual asymptotic formulas for the eigenvalues and eigenvectors of large Hermitian Toeplitz matrices generated by symbols satisfying the so-called simple-loop condition, which means that the symbol has only two intervals of monotonicity, its first derivative does not vanish on these intervals, and the second derivative is different from zero at the minimum and maximum points. Moreover, in previous works it was supposed that the symbol belongs to the weighted Wiener algebra W ^α for α ≥ 4, or satisfies even stronger smoothness conditions. We now use a different technique, which allows us to extend previous results to the case α ≥ 1 with additional smoothness at the minimum and maximum points.

J. M. Bogoya, S. M. Grudsky, E. A. Maximenko

On the Asymptotic Behavior of a Log Gas in the Bulk Scaling Limit in the Presence of a Varying External Potential II

Abstract

In this paper we continue our analysis [3] of the determinant det\( ({I}- {\gamma}{K}_{s}), \ {\gamma} \ {\in} \ (0, 1)\) where K _s is the trace class operator acting in L ²(−1, 1) with kernel \({K}_{s}(\lambda, \mu) = \frac{{\rm{sin}} \ {s}(\lambda-\mu)}{{\pi}(\lambda-\mu)}\). In [3] various key asymptotic results were stated and utilized, but without proof: Here we provide the proofs (see Theorem 1.2 and Proposition 1.3 below).

Thomas Bothner, Percy Deift, Alexander Its, Igor Krasovsky

Useful Bounds on the Extreme Eigenvalues and Vectors of Matrices for Harper’s Operators

Abstract

In analyzing a simple random walk on the Heisenberg group we encounter the problem of bounding the extreme eigenvalues of an n×n matrix of the form M = C+D where C is a circulant and D a diagonal matrix. The discrete Schrödinger operators are an interesting special case. TheWeyl and Horn bounds are not useful here. This paper develops three different approaches to getting good bounds. The first uses the geometry of the eigenspaces of C and D, applying a discrete version of the uncertainty principle. The second shows that, in a useful limit, the matrix M tends to the harmonic oscillator on L ²(ℝ) and the known eigenstructure can be transferred back. The third approach is purely probabilistic, extending M to an absorbing Markov chain and using hitting time arguments to bound the Dirichlet eigenvalues. The approaches allow generalization to other walks on other groups.

Daniel Bump, Persi Diaconis, Angela Hicks, Laurent Miclo, Harold Widom

Fast Inversion of Centrosymmetric Toeplitz-plus-Hankel Bezoutians

Abstract

This paper establishes an algorithm for the computation of the inverse of a nonsingular, centrosymmetric Toeplitz-plus-Hankel Bezoutian B of order n. The algorithm has O(n ²) computational complexity. In comparison with a previous paper on this topic the main key here is the reduction to the inversion of two symmetric Toeplitz Bezoutians of order n. This approach leads to a simpler algorithm, but it requires an additional assumption in one case. Furthermore, we obtain an explicit representation of B ⁻¹ as a sum of a Toeplitz and a Hankel matrix.

Torsten Ehrhardt, Karla Rost

On Matrix-valued Stieltjes Functions with an Emphasis on Particular Subclasses

Abstract

The paper deals with particular classes of q×q matrix-valued functions which are holomorphic in \(\mathbb{C}\backslash [\alpha, +\infty)\), where α is an arbitrary real number. These classes are generalizations of classes of holomorphic complex-valued functions studied by Kats and Krein [17] and by Krein and Nudelman [19]. The functions are closely related to truncated matricial Stieltjes problems on the interval [α+∞). Characterizations of these classes via integral representations are presented. Particular emphasis is placed on the discussion of the Moore–Penrose inverse of these matrix-valued functions.

Bernd Fritzsche, Bernd Kirstein, Conrad Mädler

The Theory of Generalized Locally Toeplitz Sequences: a Review, an Extension, and a Few Representative Applications

Abstract

We review and extend the theory of Generalized Locally Toeplitz (GLT) sequences, which goes back to Tilli’s work on Locally Toeplitz sequences and was developed by the second author during the last decade. Informally speaking, a GLT sequence {A _n } _n is a sequence of matrices with increasing size equipped with a function κ (the so-called symbol). We write {A _n } _n ~glt κ to indicate that {A _n } _n is a GLT sequence with symbol κ. This symbol characterizes the asymptotic singular value distribution of {A _n } _n ; if the matrices A _n are Hermitian, it also characterizes the asymptotic eigenvalue distribution of {A _n } _n . Three fundamental examples of GLT sequences are: (i) the sequence of Toeplitz matrices generated by a function f in L ¹; (ii) the sequence of diagonal sampling matrices containing the samples of a Riemann-integrable function a over equispaced grids; (iii) any zero-distributed sequence, i.e., any sequence of matrices with an asymptotic singular value distribution characterized by 0. The symbol of the GLT sequence (i) is f, the symbol of the GLT sequence (ii) is a, and the symbol of the GLT sequences (iii) is 0. The set of GLT sequences is a *-algebra. More precisely, suppose that {A _n ⁽ⁱ⁾ } _n ~glt κ _i for i = 1, … ,r, and let A _n = ops(A _n ⁽¹⁾ , … , A _n ^(r) ) be a matrix obtained from A _n ⁽¹⁾ , … , A _n ^(r) by means of certain algebraic operations “ops”, such as linear combinations, products, inversions and conjugate transpositions; then {A _n } _n ~glt k = ops(k ₁, … , k _r ).

The theory of GLT sequences is a powerful apparatus for computing the asymptotic singular value and eigenvalue distribution of the discretization matrices A _n arising from the numerical approximation of continuous problems, such as integral equations and, especially, partial differential equations. Indeed, when the discretization parameter n tends to infinity, the matrices A _n give rise to a sequence {A _n } _n , which often turns out to be a GLT sequence.

Nevertheless, this work is not primarily concerned with the applicative interest of the theory of GLT sequences. Although we will provide some illustrative applications at the end, the attention is focused on the mathematical foundations of the theory. We first propose a modification of the original definition of GLT sequences. With the new definition, we are able to enlarge the applicability of the theory, by generalizing/simplifying a lot of key results. In particular, we remove the Riemann-integrability assumption from the main spectral distribution and algebraic results for GLT sequences. As a final step, we extend the theory. We first prove an approximation result, which is useful to show that a given sequence of matrices is a GLT sequence. By using this result, we provide a new and easier proof of the fact that {A _n ⁻¹ } _n ~glt k ⁻¹ whenever {A _n } _n ~glt k, the matrices A _n are invertible, and κ ≠ 0 almost everywhere. Finally, using again the approximation result, we prove that {f(A _n )} _n ~glt f(k) whenever {A _n } _n ~glt k, the matrices A _n are Hermitian, and f : ℝ → ℝ is continuous.

Carlo Garoni, Stefano Serra-Capizzano

The Bézout Equation on the Right Half-plane in a Wiener Space Setting

Abstract

This paper deals with the Bézout equation \({G}(s){X}(s) = {I}_{m}, \mathfrak{R}{s} \leq {0}\), in the Wiener space of analytic matrix-valued functions on the right halfplane. In particular, G is an m × p matrix-valued analytic Wiener function, where p ≥ m, and the solution X is required to be an analyticWiener function of size p × m. The set of all solutions is described explicitly in terms of a p × p matrix-valued analyticWiener function Y , which has an inverse in the analytic Wiener space, and an associated inner function Θ defined by Y and the value of G at infinity. Among the solutions, one is identified that minimizes the H ²- norm. A Wiener space version of Tolokonnikov’s lemma plays an important role in the proofs. The results presented are natural analogs of those obtained for the discrete case in [11].

G. J. Groenewald, S. ter Horst, M. A. Kaashoek

On a Collocation-quadrature Method for the Singular Integral Equation of the Notched Half-plane Problem

Abstract

A Cauchy singular integral equation describing the notched halfplane problem of two-dimensional elasticity theory is considered. This equation contains an additional fixed singularity represented by a Mellin convolution operator. We study a polynomial collocation-quadrature method for its numerical solution which takes into account the “natural” asymptotic of the solution at the endpoints of the integration interval and for which until now no criterion for stability is known. The present paper closes this gap. For this, we present a new technique of proving that the operator sequence of the respective collocation-quadrature method belongs to a certain C ^∗-algebra, in which we can study the stability of these sequences. One of the main ingredients of this technique is to show that the part of the operator sequence, associated with the Mellin part of the original equation, is “very close” to the finite section of particular operators belonging to a C ^∗-algebra of Toeplitz operators. Moreover, basing on these stability results numerical results are presented obtained by an implementation of the proposed method.

Peter Junghanns, Robert Kaiser

The Haseman Boundary Value Problem with Slowly Oscillating Coefficients and Shifts

Abstract

The paper is devoted to studying the Haseman boundary value problem Φ⁺ ∘ α = GΦ⁻ + g on a star-like Carleson curve Γ composed by logarithmic spirals in the setting of Lebesgue spaces, where Φ^± are angular boundary values of an unknown analytic function Φ on Γ, G and g are given functions, and α is an orientation-preserving homeomorphism of Γ onto itself. This problem is reduced to the equivalent singular integral operator with a shift T = V _{α +} + GP ₋ on a Lebesgue space L ^p (Γ), where the operators P _± = 2⁻¹(I± S _Γ) are related to the Cauchy singular integral operator S _Γ, and the shift operator V _α is given by V _α f = f ∘ α. Applying the theory of Mellin pseudodifferential operators with non-regular symbols of limited smoothness and essentially decreasing the smoothness of the shift α, we establish a Fredholm criterion and an index formula for the operator T provided that the shift derivative α’ and the coefficient G are slowly oscillating functions on Γ.

Yu. I. Karlovich

On the Norm of Linear Combinations of Projections and Some Characterizations of Hilbert Spaces

Abstract

Let \( \mathcal{B} \) be a Banach space and let \( P,Q \,(P,Q\, \neq 0) \) be two complementary projections in \( \mathcal{B} \, ({\rm i.e.,}\, P \,+\,Q\,=\,I) \). For dim \( \mathcal{B} > 2 \) we show that formulas of the kind \( \parallel {aP \, +\, bQ \parallel \, = \, f(a, b, \, \parallel P\parallel)} \) hold if and only if the norm in \( \mathcal{B} \) can be induced by an inner product. The two-dimensional case needs special consideration which is done in the last two sections.

Nahum Krupnik, Alexander Markus

Pseudodifferential Operators in Weighted Hölder–Zygmund Spaces of Variable Smoothness

Abstract

We consider pseudodifferential operators of variable orders acting in Hölder–Zygmund spaces of variable smoothness. We prove the boundedness and compactness of the operators under consideration and study the Fredholm property of pseudodifferential operators with slowly oscillating at infinity symbols in the weighted Hölder–Zygmund spaces of variable smoothness.

Vadim Kryakvin, Vladimir Rabinovich

Commutator Estimates Comprising the Frobenius Norm – Looking Back and Forth

Abstract

The inequality \( {\parallel {XY-YX} \parallel}{_{F} \leq \sqrt{2}}{\parallel X \parallel}_{F}{\parallel Y\parallel}_{F} \) has some history to date. The growth of the task will be highlighted, supplemented by a look at future developments. On this way, we meet different forms and give an insight into various consequences of it. The collection of results will be enriched by introductive explanations. We also cross other fields that are important for theory and applications, and even uncover less known relationships.

Zhiqin Lu, David Wenzel

Numerical Ranges of 4-by-4 Nilpotent Matrices: Flat Portions on the Boundary

Abstract

In their 2008 paper Gau and Wu conjectured that the numerical range of a 4-by-4 nilpotent matrix has at most two flat portions on its boundary. We prove this conjecture, establishing along the way some additional facts of independent interest. In particular, a full description of the case in which these two portions indeed materialize and are parallel to each other is included.

Erin Militzer, Linda J. Patton, Ilya M. Spitkovsky, Ming-Cheng Tsai

Traces on Operator Ideals and Related Linear Forms on Sequence Ideals (Part IV)

Abstract

The concept of a dyadic representation was used for the first time in 1963, when I constructed traces of operators (acting on a Banach space) whose sequence of approximation numbers is summable. Only recently, in a series of papers, those representations played the decisive role in describing all traces on arbitrary operator ideals over the separable infinite-dimensional Hilbert space. Now this method is extended to operator ideals on Banach spaces defined by means of generalized approximation numbers. The results are demonstrated on the example of convolution operators generated by functions belonging to certain Lipschitz/Besov spaces.

Albrecht Pietsch

Error Estimates for the ESPRIT Algorithm

Abstract

Let \( z_{j} \,:=e^{fj}\, (j=1,\ldots,M) \) with \( f_{j} \, \epsilon \, [-\varphi, \,0] + {\rm i}[-\pi, \,\pi) \) and small φ ≥ 0 be distinct nodes. With complex coefficients \( c_{j} \neq \,0 \), we consider an exponential sum \( h(x)\, :=c_{1} \,e^{f_{1}\, x} + \,\ldots \, + c_{M}\, e^{f_{M} \, x} (x \geq 0) \). Many applications in electrical engineering, signal processing, and mathematical physics lead to the following problem: Determine all parameters of h, if N noisy sampled values \( \tilde{h}_{k} \,:=h(k) + e_{k}\, (k = 0,\ldots,N-1) \) with N ≫ 2M are given, where ek are small error terms. This parameter identification problem is a nonlinear inverse problem which can be efficiently solved by the ESPRIT algorithm. In this paper, we present mainly corresponding error estimates for the nodes \( z_{j}(j=1,\ldots,M) \). We show that under appropriate conditions, the results of the ESPRIT algorithm are relatively insensitive to small perturbations on the sampled data.

Daniel Potts, Manfred Tasche

The Universal Algebra Generated by a Power Partial Isometry

Abstract

The goal of this paper is to characterize (a slight modification of) the algebra of the finite sections method for Toeplitz operators with continuous generating function, as first described by Albrecht Böttcher and Bernd Silbermann in [2], by a universal property, namely as the universal C ^*-algebra generated by a power partial isometry (PPI). A PPI is an element of a C ^*-algebra with the property that every non-negative power of that element is a partial isometry.

Steffen Roch

Norms, Condition Numbers and Pseudospectra of Convolution Type Operators on Intervals

Abstract

The results in this paper describe the asymptotic behavior of convolution type operators on finite intervals as the length of these intervals tends to infinity. The family of operators under consideration here is generated (among others) by Fourier convolutions with slowly oscillating, almost periodic, bounded and uniformly continuous, and quasi-continuous multipliers, as well as operators of multiplication by slowly oscillating, almost periodic, and piecewise continuous functions. The focus is on the convergence of norms, condition numbers and pseudospectra.

Markus Seidel

Paired Operators in Asymmetric Space Setting

Abstract

Relations between paired and truncated operators acting in Banach spaces are generalized to asymmetric space settings, i.e., to matrix operators acting between different spaces. This allows more direct proofs and further results in factorization theory, here in connection with the Cross Factorization Theorem and the Bart–Tsekanovsky Theorem. Concrete examples from mathematical physics are presented: the construction of resolvent operators to problems of diffraction of time-harmonic waves from plane screens which are not convex.

Frank-Olme Speck

Natural Boundary for a Sum Involving Toeplitz Determinants

Abstract

In the theory of the two-dimensional Ising model, the diagonal susceptibility is equal to a sum involving Toeplitz determinants. In terms of a parameter k the diagonal susceptibility is analytic for |k| < 1, and the authors proved the conjecture that this function has the unit circle as a natural boundary. The symbol of the Toepltiz determinants was a k-deformation of one with a single singularity on the unit circle. Here we extend the result, first, to deformations of a larger class of symbols with a single singularity on the unit circle, and then to deformations of (almost) general Fisher–Hartwig symbols.

Craig A. Tracy, Harold Widom

A Riemann–Hilbert Approach to Filter Design

Abstract

The paper is devoted to interrelations between boundary value problems of Riemann–Hilbert type and optimization problems in spaces of bounded holomorphic functions which are motivated by optimal filter design. A numerical method of Newton type for the iterative solution of nonlinear Riemann–Hilbert problems is adapted for solving the optimization problem and its convergence is proved. The approach is illustrated by the design of a low-pass filter, which we discuss in some detail.

My long-time acquaintance with Albrecht Böttcher, and in particular our journey to a conference on H ^∞ control at lake Como in 1990, form the background of the paper.

Elias Wegert