Marko Lindner

Technische Universität Hamburg | TUHH · Institute of Mathematics

An operator on an $l^{p}$-space is called band-dominated if it can be approximated, in the operator norm, by operators with a banded matrix representation. It is known that a rich band-dominated operator is $\mathcal{P}$-Fredholm (which is a generalization of the classical Fredholm property) if and only if all of its so-called limit operators are i...

We study the spectra and pseudospectra of finite and infinite tridiagonal random matrices, in the case where each of the diagonals varies over a separate compact set, say $U,V,W\subset\mathbb{C}$. Such matrices are sometimes termed stochastic Toeplitz matrices $A_+$ in the semi-infinite case and stochastic Laurent matrices $A$ in the bi-infinite ca...

An operator $A$ on an $l^p$-space is called band-dominated if it can be approximated, in the operator norm, by operators with a banded matrix representation. The coset of $A$ in the Calkin algebra determines, for example, the Fredholmness of $A$, the Fredholm index, the essential spectrum, the essential norm and the so-called essential pseudospectr...

By counting 1's in the "right half" of $2w$ consecutive rows, we locate the main diagonal of any doubly infinite permutation matrix with bandwidth $w$. Then the matrix can be correctly centered and factored into block-diagonal permutation matrices. Part II of the paper discusses the same questions for the much larger class of band-dominated matrice...

In the first half of this text we explore the interrelationships between the abstract theory of limit operators (see e.g. the recent monographs of Rabinovich, Roch & Silbermann and Lindner) and the concepts and results of the generalised collectively compact operator theory introduced by Chandler-Wilde and Zhang. We build up to results obtained by...

We obtain sequences of inclusion sets for the spectrum, essential spectrum, and pseudospectrum of banded, in general non-normal, matrices of finite or infinite size. Each inclusion set is the union of the pseudospectra of certain submatrices of a chosen size $n$. Via the choice of $n$, one can balance accuracy of approximation against computational...

For a bounded linear operator on a Banach space, we study approximation of the spectrum and pseudospectra in the Hausdorff distance. We give sufficient and necessary conditions in terms of pointwise convergence of appropriate spectral quantities.

We demonstrate criteria, purely based on finite subwords of the potential, to guarantee spectral inclusion as well as Hausdorff approximation of pseudospectra or even spectra of generalized Schr\"odinger operators on the discrete line or half-line. In fact, our results are neither limited to Schr\"odinger or self-adjoint operators, nor to Hilbert s...

We study 1D discrete Schr\"odinger operators $H$ with integer-valued potential and show that, $(i)$, invertibility (in fact, even just Fredholmness) of $H$ always implies invertibility of its half-line compression $H_+$ (zero Dirichlet boundary condition, i.e. matrix truncation). In particular, the Dirichlet eigenvalues avoid zero -- and all other...

For a bounded linear operator on a Banach space, we study approximation of the spectrum and pseudospectra in the Hausdorff distance. We give sufficient and necessary conditions in terms of pointwise convergence of appropriate spectral quantities.

We study two abstract scenarios, where an operator family has a certain minimality property. In both scenarios, it is shown that norm, spectrum and resolvent are the same for all family members. Both abstract settings are illustrated by practically relevant examples, including discrete Schr\"odinger operators with periodic, quasiperiodic, almost-pe...

We study discrete Schr\"odinger operators $H$ with periodic potentials as they are typically used to approximate aperiodic Schr\"odinger operators like the Fibonacci Hamiltonian. We prove an efficient test for applicability of the finite section method, a procedure that approximates $H$ by growing finite square submatrices $H_n$. For integer-valued...

For the study of electromagnetic waves in a 2D environment, the Helmholtz equation can be reformulated in boundary integral form by the so‐called Contour Integral Method (CIM). The influence of uncertainty in the geometry is studied via Polynomial Chaos Expansion (PCE). A distinction is made between the so‐called intrusive and non‐intrusive PCE. Bo...

This paper presents a multiscale method for the numerically efficient electromagnetic analysis of two-dimensional photonic and electromagnetic crystals. It is based on a contour integral method and a segmented analysis of more complex structures in terms of building blocks which are models for essential components. The scattering properties of esse...

We review approximation methods and their stability and applicability. We then focus on the finite section method and Galerkin methods and show that on separable Hilbert spaces either one can be interpreted as the other. In the end we demonstrate that well‐known methods such as the finite element method and polynomial chaos expansion are particular...

We study finite but growing principal square submatrices An of the one- or two-sided infinite Fibonacci Hamiltonian A. Our results show that such a sequence (An), no matter how the points of truncation are chosen, is always stable – implying that An is invertible for sufficiently large n and A–1n → A–1 pointwise. © 2018, Springer International Publ...

In their recent paper "The spectral norm of a Horadam circulant matrix", Merikoski, Haukkanen, Mattila and Tossavainen study under which conditions the spectral norm of a general real circulant matrix ${\bf C}$ equals the modulus of its row/column sum. We improve on their sufficient condition until we have a necessary one. Our results connect the a...

It is well known that, given an equivariant and continuous (in a suitable sense) family of selfadjoint operators in a Hilbert space over a minimal dynamical system, the spectrum of all operators from that family coincides. As shown recently similar results also hold for suitable families of non-selfadjoint operators in $\ell^p (\ZM)$. Here, we gene...

We study finite but growing principal square submatrices $A_n$ of the one- or two-sided infinite Fibonacci Hamiltonian $A$. Our results show that such a sequence $(A_n)$, no matter how the points of truncation are chosen, is always stable -- implying that $A_n$ is invertible for sufficiently large $n$ and $A_n^{-1}\to A^{-1}$ pointwise.

We study spectra and pseudospectra of certain bounded linear operators on \ell^2(Z). The operators are generally non-normal, and their matrix representation has a characteristic off-diagonal decay. Based on a result of Chandler-Wilde, Chonchaiya and Lindner for tridiagonal infinite matrices, we demonstrate an efficient algorithm for the computation...

We study spectra and pseudospectra of certain bounded linear operators on $\ell^2({\mathbb Z})$. The operators are generally non-normal, and their matrix representation has a characteristic off-diagonal decay. Based on a result of Chandler-Wilde, Chonchaiya and Lindner for tridiagonal infinite matrices, we demonstrate an efficient algorithm for the...

The study of spectral properties of linear operators on an infinite-dimensional Hilbert space is of great interest. This task is especially difficult when the operator is non-selfadjoint or even non-normal. Standard approaches like spectral approximation by finite sections generally fail in that case. In this talk we present an algorithm which rigo...

We consider equivariant continuous families of discrete one-dimensional operators over arbitrary dynamical systems. We introduce the concept of a pseudo-ergodic element of a dynamical system. We then show that all operators associated to pseudo-ergodic elements have the same spectrum and that this spectrum agrees with their essential spectrum. As a...

In this paper we develop and apply methods for the spectral analysis of non-self-adjoint tridiagonal infinite and finite random matrices, and for the spectral analysis of analogous deterministic matrices which are pseudo-ergodic in the sense of E.B.Davies (Commun. Math. Phys. 216 (2001), 687-704). As a major application to illustrate our methods we...

The purpose of this paper is to prove that the spectrum of the non-self-adjoint one-particle Hamiltonian proposed by J. Feinberg and A. Zee (Phys. Rev. E 59 (1999), 6433--6443) has interior points. We do this by first recalling that the spectrum of this random operator is the union of the set of $\ell^\infty$ eigenvalues of all infinite matrices wi...

We consider the classical coupled, combined-field integral equation formulations for time-harmonic acoustic scattering by a sound soft bounded obstacle. In recent work, we have proved lower and upper bounds on the $L^2$ condition numbers for these formulations, and also on the norms of the classical acoustic single- and double-layer potential opera...

This article is about a problem in the numerical analysis of random operators. We study a version of the finite section method for the approximate solution of equations $Ax=b$ in infinitely many variables, where $A$ is a random Jacobi operator. In other words, we approximately solve infinite second order difference equations with stochastic coeffic...

In this article we compare the set of integer points in the homothetic copy \({n\Pi}\) of a lattice polytope \({\Pi\subseteq{{\mathbb R}}^d}\) with the set of all sums x 1 + . . . + x n with \({x_1,\ldots,x_n\in \Pi\cap{{\mathbb Z}}^d}\) and \({n\in{{\mathbb N}}}\) . We give conditions on the polytope \({\Pi}\) under which these two sets coincide a...

In the first part of this paper we give an elementary proof of the fact that if an infinite matrix $A$, which is invertible as a bounded operator on $\ell^2$, can be uniformly approximated by banded matrices then so can the inverse of $A$. We give explicit formulas for the banded approximations of $A^{-1}$ as well as bounds on their accuracy and sp...

In this article we demonstrate and compare two modified versions of the classical finite section method for band-dominated operators in case the latter is not stable. For both methods we give explicit criteria for their applicability.

This text is concerned with the Fredholm theory and stable approximation of bounded linear operators generated by a class of infinite matrices $(a_{ij})$ that are either banded or have certain decay properties as one goes away from the main diagonal. The operators are studied on $\ell^p$ spaces of functions $\Z^N\to X$, where $p\in[1,\infty]$, $N\i...

This text is concerned with the Fredholm theory and stable approximation of bounded linear operators generated by a class of infinite matrices $(a_{ij})$ that are either banded or have certain decay properties as one goes away from the main diagonal. The operators are studied on $\ell^p$ spaces of functions $\Z^N\to X$, where $p\in[1,\infty]$, $N\i...

We study the classical combined field integral equation formulations for time-harmonic acoustic scattering by a sound soft bounded obstacle, namely the indirect formulation due to Brakhage-Werner/Leis/Panic, and the direct formulation associated with the names of Burton and Miller. We obtain lower and upper bounds on the condition numbers for these...

The purpose of this note is to prove a sufficient and necessary criterion on the stability of a subsequence of the finite section method for a so-called band-dominated operator on p (Z N , X). We hereby generalize previous results into several directions: We generalize the subsequence theorem from dimension N = 1 (see [11]) to arbitrary dimensions...

The purpose of this note is to demonstrate the use of the results from [5, 6] for the explicit computation of the spectrum of two-sided infinite matrices with random diagonals. Here we consider the case of two random diagonals, one of them the main diagonal. Our result is a generalization of [24, Theorem 8.1] by Trefethen, Contedini and Embree from...

We consider a class of boundary integral equations that arise in the study of strongly elliptic BVPs in unbounded domains of the form $D = \{(x, z)\in \mathbb{R}^{n+1} : x\in \mathbb{R}^n, z > f(x)\}$ where $f : \mathbb{R}^n \to\mathbb{R}$ is a sufficiently smooth bounded and continuous function. A number of specific problems of this type, for exam...

The purpose of this paper is to show that, for a large class of band-dominated operators on $\ell^\infty(Z,U)$, with $U$ being a complex Banach space, the injectivity of all limit operators of $A$ already implies their invertibility and the uniform boundedness of their inverses. The latter property is known to be equivalent to the invertibility at...

We introduce a novel multi-section method for the solution of integral equa- tions on unbounded domains. The method is applied to the rough-surface scattering problem in three dimensions, in particular to a Brakhage-Werner type integral equation for acoustic scattering by an unbounded rough surface with Dirichlet boundary condition, where the funda...

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The purpose of this paper is to demonstrate the so-called Fredholm-inverse closedness of the Wiener algebra W and to deduce independence of the Fredholm property and index of the underlying space. More precisely, we look at operators A ∈ W as acting on a family of vector valued p spaces and show that the Fredholm regularizer of A for one of these s...

In this book we are concerned with the study of a certain class of in?nite matrices and two important properties of them: their Fredholmness and the stability of the approximation by their ?nite truncations. Let us take these two properties as a starting point for the big picture that shall be presented in what follows. Stability Fredholmness We th...

The purpose of this note is to show that the finite section method for a band operator with slowly oscillating coefficients is stable if and only if the operator is invertible. This result generalizes the classical stability criterion for the finite section method for band Toeplitz operators (= the case of constant coefficients).

Limit operators have proven to be a device for the study of several properties of an operator including Fredholmness and invertibility at infinity, but also the applicability of approximation methods. For band-dominated operators, the question of existence and structure of their limit operators essentially reduces to the study of multiplication ope...

The topic of this paper is band operators and the norm limits of such — so-called band-dominated operators, both classes acting on L∞(ℝn). Invertibility at infinity is closely related to Fredholmness. In fact, in the discrete case ℓp(ℤn), 1 ≤ p ≤ ∞, both properties coincide. For many applications, e.g., the question of applicability of certain appr...

We present an approach to the finite section method for band-dominated operators—the norm-limits of band operators on . We hereby show that the sequence of finite sections is stable if and only if some associated operator is invertible at infinity. By means of the theory in Lindner and Silbermann (Lindner, M., Silbermann, B. (20037. Lindner , M. a...

Die Dissertation untersucht die Invertierbarkeit im Unendlichen fuer Normgrenzwerte von Bandoperatoren - sogenannte band-dominierte Operatoren. Das dazu verwendete Instrument ist die Methode der Limitoperatoren. Es werden grundlegende Eigenschaften von Limitoperatoren bewiesen, Zusammenhaenge zur Invertierbarkeit im Unendlichen hergeleitet, sowie d...

Die Dissertation untersucht die Invertierbarkeit im Unendlichen fuer Normgrenzwerte von Bandoperatoren - sogenannte band-dominierte Operatoren. Das dazu verwendete Instrument ist die Methode der Limitoperatoren. Es werden grundlegende Eigenschaften von Limitoperatoren bewiesen, Zusammenhaenge zur Invertierbarkeit im Unendlichen hergeleitet, sowie d...

This paper explores the relationship between the spectra of perturbed infinite banded Laurent matricesL(a)+K and their approximations by perturbed circulant matricesC n (a)+P n KP n for largen. The entriesK jk of the perturbation matrices assume values in prescribed sets limn ® ¥ ÈK Î KW E \textsp \text (Cn (a) + Pn KPn ) = ÈK Î KW E \textsp \tex...

This book is concerned with the study of infinite matrices and their approximation by matrices of finite size. The framework includes both the simplest, important case where the matrix entries are numbers, and the more general case where the entries are bounded linear operators. This generality ensures that examples of the class of operators studie...