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Convex Analysisの二,三の進展について

The fundamental theorems on the asymptotic behavior of eigenvalues, inverses, and products of banded Toeplitz matrices and Toeplitz matrices with absolutely summable elements are derived in a tutorial manner. Mathematical elegance and generality are sacrificed for conceptual simplicity and insight in the hope of making these results available to engineers lacking either the background or endurance to attack the mathematical literature on the subject. By limiting the generality of the matrices considered, the essential ideas and results can be conveyed in a more intuitive manner without the mathematical machinery required for the most general cases. As an application the results are applied to the study of the covariance matrices and their factors of linear models of discrete time random processes.

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Toeplitz and circulant matrices

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

Physical interactions in quantum many-body systems are typically local: Individual constituents interact mainly with their few nearest neighbors. This locality of interactions is inherited by a decay of correlation functions, but also reflected by scaling laws of a quite profound quantity: the entanglement entropy of ground states. This entropy of the reduced state of a subregion often merely grows like the boundary area of the subregion, and not like its volume, in sharp contrast with an expected extensive behavior. Such ``area laws'' for the entanglement entropy and related quantities have received considerable attention in recent years. They emerge in several seemingly unrelated fields, in the context of black hole physics, quantum information science, and quantum many-body physics where they have important implications on the numerical simulation of lattice models. In this Colloquium the current status of area laws in these fields is reviewed. Center stage is taken by rigorous results on lattice models in one and higher spatial dimensions. The differences and similarities between bosonic and fermionic models are stressed, area laws are related to the velocity of information propagation in quantum lattice models, and disordered systems, nonequilibrium situations, and topological entanglement entropies are discussed. These questions are considered in classical and quantum systems, in their ground and thermal states, for a variety of correlation measures. A significant proportion is devoted to the clear and quantitative connection between the entanglement content of states and the possibility of their efficient numerical simulation. Matrix-product states, higher-dimensional analogs, and variational sets from entanglement renormalization are also discussed and the paper is concluded by highlighting the implications of area laws on quantifying the effective degrees of freedom that need to be considered in simulations of quantum states.

Colloquium: Area laws for the entanglement entropy

Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

This is an essay containing personal reminiscences and describing selected topics of joint mathematical work of the author and Sergei Grudsky. The topics have their focus on Toeplitz operators and large finite Toeplitz matrices.

The part of my path I walked together with Sergei Grudsky

https://ems.press/content/serial-article-files/8990

Fredholmness and finite section method for Toeplitz operators in $l^p (Z_+ \times Z_+)$ with piecewise continuous symbols I

In this chapter our leading actor, the Cauchy singular integral operator S, enters the scene. A very deep theorem, which should actually be Theorem 1 of this book, says that S is bounded on L P (Γ, ω) (1 <p < ∞) if and only if Γ is a Carleson curve and ω is a Muckenhoupt weight in A P (Γ). The proof of this theorem is difficult and goes beyond the scope of this book. We nevertheless decided to write down a proof, but this proof will only be given in Chapter 5. The purpose of this chapter is to provide the reader with some facts and results that should suffice to understand Chapters 6 to 10 without browsing in Chapter 5.

Boundedness of the Cauchy singular integral

This paper is motivated by recent studies of Huang et al. on distributed PCA and network anomaly detection and contains a rigorous derivation of bounds for the expected value and the variance of the spectral norm of the error in large covariance matrices. This derivation is based on a deep result by Yin et al. (Probab. Theor. Relat. Fields 1988; 78:509–521), which gives the asymptotics of the maximal eigenvalue of a random matrix as the matrix dimension goes to infinity. Copyright © 2007 John Wiley & Sons, Ltd.

Albrecht Böttcher

Papers

The part of my path I walked together with Sergei Grudsky

Fredholmness and finite section method for Toeplitz operators in $l^p (Z_+ \times Z_+)$ with piecewise continuous symbols I

Boundedness of the Cauchy singular integral

Rigorous stochastic bounds for the error in large covariance matrices

Twenty old and new asymptotic results on Toeplitz band matrices