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Albrecht Böttcher

Publications and source records attributed to Albrecht Böttcher.

6 recordsLinked to original sources
arXiv2026

Condition numbers of block Toeplitz matrices and stability of space-time IgA approximations for the wave and Schrödinger equations

In previous work by several authors, the behavior of the condition numbers of banded Toeplitz matrices was studied as the matrix size tends to infinity. In the present contribution, two main directions are pursued. As a first step, we extend this study to block Toeplitz matrices with blocks of fixed size $N$. As in the scalar case, we show that even when the symbol generates a Fredholm infinite Toeplitz operator, the condition numbers of the finite matrices may grow at least exponentially. Upper and lower bounds for the condition numbers are obtained, and examples showing that they may grow arbitrarily fast are presented. Then, as a second step, we apply the developed theory to the stability analysis of space-time Galerkin methods, where in time an Isogeometric approach is used with regularity $r$, $1\le r\le p-1$, $p$ being the employed polynomial degree. These stability issues are related exactly to the conditioning of block Toeplitz-like matrices with blocks of size $N=p-r$. Specific examples are treated in detail and related numerical experiments are presented and critically discussed. We finally present a short list of relevant open problems.

math.NA↗
arXiv2026

Uncertainty principles and lower bounds for Schrödinger operators

We prove that two different abstract quantitative uncertainty principles are equivalent to the strict positivity of the associated abstract Schrö\-din\-ger operators. We also discuss the case of continuum Schrödinger operators, in which case our method provides an explicitly computable lower bound as well as a control theoretic application.

math.SP↗
arXiv2026

The wanted extension of Fujii and Tsurumaru's formula for the spectral radius of the Bell-CHSH operator

This paper is motivated by a recent paper of Yuki Fujii and Toyohiro Tsurumaru in which they established a beautiful formula for the spectral radius of the Bell-CHSH operator on finite-dimensional Hilbert spaces. To tackle the operator on infinite-dimensional spaces, they elaborated a method based on appropriate approximation of commutators of infinite-dimensional orthogonal projections by commutators of orthogonal projections on finite-dimensional spaces. We here give a proof of Fujii and Tsurumaru's original formula that works in all dimensions. We also present an alternative approximation procedure, uncover the connection of the problem with block Toeplitz operators, and derive good estimates and explicit expressions for the spectral radius in concrete cases.

math.FA↗
arXiv2021

The norm attainment problem for functions of projections

The paper is concerned with the problem of identifying the norm attaining operators in the von Neumann algebra generated by two orthogonal projections on a Hilbert space. This algebra contains every skew projection on that Hilbert space and hence the results of the paper also describe functions of skew projections and their adjoints that attain the norm.

math.FA↗
arXiv2006

Generalized Krein algebras and asymptotics of Toeplitz determinants

We give a survey on generalized Krein algebras $K_{p,q}^{α,β}$ and their applications to Toeplitz determinants. Our methods originated in a paper by Mark Krein of 1966, where he showed that $K_{2,2}^{1/2,1/2}$ is a Banach algebra. Subsequently, Widom proved the strong Szegő limit theorem for block Toeplitz determinants with symbols in $(K_{2,2}^{1/2,1/2})_{N\times N}$ and later two of the authors studied symbols in the generalized Krein algebras $(K_{p,q}^{α,β})_{N\times N}$, where $λ:=1/p+1/q=α+β$ and $λ=1$. We here extend these results to $0<λ<1$. The entire paper is based on fundamental work by Mark Krein, ranging from operator ideals through Toeplitz operators up to Wiener-Hopf factorization.

math.FA↗