arXiv subjects

Marko Lindner

Publications and source records attributed to Marko Lindner.

At least 19 recordsLinked to original sources

arXiv2026

Localisation of pseudospectra on discrete groups

In this paper we generalise two of the methods and corresponding results from our previous paper ``On spectral inclusion sets and computing the spectra and pseudospectra of bounded linear operators'' [J. Spectr. Theory 14 (2024), 719--804] from tridiagonal operators on $\ell^2(\Z)$ to band operators $A$ on $\ell^2(G,Y)$ with a countable Abelian group $G$ and a Hilbert space $Y$. Again, we cover the pseudospectra of $A$, with error-control, via a union of pseudospectra of finite and moderately sized ``local patches'' of $A$. While a major application is to understand the case $G=\Z^d$ that is immanent in many physical problems, our new approach to the so-called $\tau$ and $\tau_1$ methods immediately extends to countable Abelian groups $G$.

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arXiv2024

Gershgorin-Type Spectral Inclusions for Matrices

In this paper we derive sequences of Gershgorin-type inclusion sets for the spectra and pseudospectra of finite matrices. In common with previous generalisations of the classical Gershgorin bound for the spectrum, our inclusion sets are based on a block decomposition. In contrast to previous generalisations that treat the matrix as a perturbation of a block-diagonal submatrix, our arguments treat the matrix as a perturbation of a block-tridiagonal matrix, which can lead to sharp spectral bounds, as we show for the example of large Toeplitz matrices. Our inclusion sets, which take the form of unions of pseudospectra of square or rectangular submatrices, build on our own recent work on inclusion sets for bi-infinite matrices [Chandler-Wilde, Chonchaiya, Lindner, {\em J. Spectr. Theory} {\bf 14}, 719--804 (2024)].

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arXiv2024

On Spectral Inclusion Sets and Computing the Spectra and Pseudospectra of Bounded Linear Operators

In this paper we derive novel families of inclusion sets for the spectrum and pseudospectrum of large classes of bounded linear operators, and establish convergence of particular sequences of these inclusion sets to the spectrum or pseudospectrum, as appropriate. Our results apply, in particular, to bounded linear operators on a separable Hilbert space that, with respect to some orthonormal basis, have a representation as a bi-infinite matrix that is banded or band-dominated. More generally, our results apply in cases where the matrix entries themselves are bounded linear operators on some Banach space. In the scalar matrix entry case we show that our methods, given the input information we assume, lead to a sequence of approximations to the spectrum, each element of which can be computed in finitely many arithmetic operations, so that, with our assumed inputs, the problem of determining the spectrum of a band-dominated operator has solvability complexity index one, in the sense of Ben-Artzi et al. (C. R. Acad. Sci. Paris, Ser. I 353 (2015), 931-936). As a concrete and substantial application, we apply our methods to the determination of the spectra of non-self-adjoint bi-infinite tridiagonal matrices that are pseudoergodic in the sense of Davies (Commun. Math. Phys. 216 (2001) 687-704).

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arXiv2023

Finite sections: stability, spectral pollution and asymptotics of condition numbers and pseudospectra

The stability of an approximating sequence $(A_n)$ for an operator $A$ usually requires, besides invertibility of $A$, the invertibility of further operators, say $B, C, \dots$, that are well-associated to the sequence $(A_n)$. We study this set, $\{A,B,C,\dots\}$, of so-called stability indicators of $(A_n)$ and connect it to the asymptotics of $\|A_n\|$, $\|A_n^{-1}\|$ and $\kappa(A_n)=\|A_n\|\|A_n^{-1}\|$ as well as to spectral pollution by showing that $\limsup {\rm Spec}_\varepsilon A_n= {\rm Spec}_\varepsilon A\cup{\rm Spec}_\varepsilon B\cup{\rm Spec}_\varepsilon C\cup\dots$. We further specify, for each of $\|A_n\|$, $\|A_n^{-1}\|$, $\kappa(A_n)$ and ${\rm Spec}_\varepsilon A_n$, under which conditions even convergence applies.

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arXiv2023

Convergent spectral inclusion sets for banded matrices

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 cost, and we show, in the case of infinite matrices, convergence as $n\to\infty$ of the respective inclusion set to the corresponding spectral set.

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arXiv2022

Spectral approximation of generalized Schr\"odinger operators via approximation of subwords

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 space or 1D.

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arXiv2022

Half-line compressions and finite sections of discrete Schr\"odinger operators with integer-valued potentials

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 integers. We use this result to conclude that, $(ii)$, the finite section method (approximate inversion via finite and growing matrix truncations) is applicable to $H$ as soon as $H$ is invertible. The same holds for $H_+$.

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arXiv2022

A note on Hausdorff-convergence of pseudospectra

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.

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arXiv2021

Minimal Families of Limit Operators

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-periodic, Sturmian and pseudo-ergodic potential. The main tool is the method of limit operators, known from studies of Fredholm operators and convergence of projection methods. We close by connecting this tool to the study of subwords of the operator potential.

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arXiv2021

Finite Sections of Periodic Schr\"odinger Operators

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 potentials, we show that the finite section method is applicable as soon as $H$ is invertible. This statement remains true for $\{0, \lambda\}$-valued potentials with fixed rational $\lambda$ and period less than nine as well as for arbitrary real-valued potentials of period two.

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arXiv2021

Finite section method for aperiodic Schr\"odinger operators

We consider 1D discrete Schr\"odinger operators with aperiodic potentials given by a Sturmian word, which is a natural generalisation of the Fibonacci Hamiltonian. Via a standard approximation by periodic potentials, we establish Hausdorff convergence of the corresponding spectra for the Schr\"odinger operators on the axis as well as for their compressions to the half-axis. Based on the half-axis results, we study the finite section method, which is another operator approximation, now by compressions to finite but growing intervals, that is often used to solve operator equations approximately. We find that, also for this purpose, the aperiodic case can be studied via its periodic approximants. Our results on the finite section method of the aperiodic operator are illustrated by confirming a result on the finite sections of the special case of the Fibonacci Hamiltonian.

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arXiv2018

Circulant matrices: norm, powers, and positivity

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 above problem to positivity of sufficiently high powers of the matrix ${\bf C^\top C}$. We then generalize the result to complex circulant matrices.

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arXiv2017

Finite sections of the Fibonacci Hamiltonian

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.

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arXiv2016

On the spectrum of operator families on discrete groups over minimal dynamical systems

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 generalize this to a large class of bounded linear operator families on Banach-space valued $\ell^p$-spaces over countable discrete groups. We also provide equality of the pseudospectra for operators in such a family. A main tool for our analysis are techniques from limit operator theory.

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arXiv2016

Recycling Givens rotations for the efficient approximation of pseudospectra of band-dominated operators

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 computation of upper and lower bounds on the pseudospectrum of operators that are merely norm limits of band matrices -- the so-called band-dominated operators. After approximation by a band matrix and fixing a parameter $n\in{\mathbb N}$, one looks at $n$ consecutive columns $\{k+1,...,k+n\}$, $k\in{\mathbb Z}$, of the corresponding matrix and computes the smallest singular value of that section via QR factorization. We here propose a QR factorization by a sequence of Givens rotations in such a way that a large part of the computation can be reused for the factorization of the next submatrix -- when $k$ is replaced by $k+1$. The computational cost for the next factorization(s) is ${\mathcal O}(nd)$ as opposed to a naive implementation with ${\mathcal O}(nd^2)$, where $d$ is the bandwidth. So our algorithm pays off for large bands, which is attractive when approximating band-dominated operators with a full (i.e. not banded) matrix.

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arXiv2015

Coburn's Lemma and the Finite Section Method for Random Jacobi Operators

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 case. Their spectra, $\Sigma=$ spec $A$ and $\Sigma_+=$ spec $A_+$, are independent of $A$ and $A_+$ as long as $A$ and $A_+$ are pseudoergodic (in the sense of E.B. Davies, Commun. Math. Phys., 2001), which holds almost surely in the random case. This was shown in Davies (2001) for $A$; that the same holds for $A_+$ is one main result of this paper. We give upper and lower bounds on $\Sigma$ and $\Sigma_+$, and we explicitly compute a set $G$ that fills the gap between the two in the sense that $\Sigma\cup G=\Sigma_+$. We show that invertibility of one operator $A_+$ implies invertibility - and uniform boundedness of the inverses - of all finite square matrices with three diagonals in $U, V$ and $W$. This implies that the so-called finite section method for the approximate solution of a system $A_+x=b$ is applicable as soon as $A_+$ is invertible, and that the same method for estimating the spectrum of $A_+$ does not suffer from spectral pollution. Both results illustrate that tridiagonal stochastic Toeplitz operators share important properties of (classical) Toeplitz operators. One of our main tools is a new version of the Coburn lemma for classical Toeplitz operators, saying that a stochastic tridiagonal Toeplitz operator, if Fredholm, is always injective or surjective. In the final part we bound and compare the norms, and the norms of inverses, of bi-infinite, semi-infinite and finite tridiagonal matrices over $U$, $V$ and $W$. This allows the study of the resolvent norms, and hence the pseudospectra, of these operators and matrices.

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arXiv2015

Essential pseudospectra and essential norms of band-dominated operators

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 pseudospectrum of $A$. This coset can be identified with the collection of all so-called limit operators of $A$. It is known that this identification preserves invertibility (hence spectra). We now show that it also preserves norms and in particular resolvent norms (hence pseudospectra). In fact we work with a generalization of the ideal of compact operators, so-called $\mathcal{P}$-compact operators, allowing for a more flexible framework that naturally extends to $l^p$-spaces with $p\in\{1,\infty\}$ and/or vector-valued $l^p$-spaces.

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arXiv2014

Note on spectra of non-selfadjoint operators over dynamical systems

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 consequence we obtain that the spectrum is constant and agrees with the essential spectrum for all elements in the dynamical system if minimality holds.

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