/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

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.

/pdf/toeplitz-and-circulant-matrices-3e6lj1a9jc.pdf

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)

The pseudospectra of banded finite dimensional Toeplitz matrices rapidly converge to the pseudospectra of the corresponding infinite dimensional operator. This exponential convergence makes a compelling case for analyzing pseudospectra of such Toeplitz matrices---not just eigenvalues. What if the matrix is dense and its symbol has a jump discontinuity? The pseudospectra of the finite matrices still converge, but it is shown here that the rate is no longer exponential in the matrix dimension---only algebraic.

/pdf/piecewise-continuous-toeplitz-matrices-and-operators-slow-2wuk3wazkc.pdf

Piecewise Continuous Toeplitz Matrices and Operators: Slow Approach to Infinity

If the symbol of an infinite Toeplitz band matrix has nonzero winding number, then the distances of the n × n truncations of this matrix to the nearest singular matrices go to zero with exponential speed as n → ∞. The purpose of this note is to show that if we measure the distance to the nearest singular matrix within the matrices of the same Toeplitz band structure as the original matrix, then things are less dramatic: this distance stays away from zero if only the origin does not belong to the Schmidt-Spitzer set.

On the Distance of a Large Toeplitz Band Matrix to the Nearest Singular Matrix

On certain finite-dimensional algebras generated by two idempotents

We establish a criterion for the applicability of the finite section method to Toeplitz operators with continuous matrix-symbols on the Bergman space of the polydisk.

Albrecht Böttcher

Papers

Piecewise Continuous Toeplitz Matrices and Operators: Slow Approach to Infinity

On the Distance of a Large Toeplitz Band Matrix to the Nearest Singular Matrix

On certain finite-dimensional algebras generated by two idempotents

Truncated Toeplitz operators on the polydisk

Local spectra of approximate identities, cluster sets and Toeplitz operators