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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)

Toeplitz Operators with PQC Symbols on Weighted Hardy Spaces

The C*-algebra of singular integral operators with semi-almost periodic coefficients

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Discrete Wiener-Hopf operators on spaces with Muckenhoupt weight

It is well known that amplitude modulation does not affect Fredholmness of Toeplitz operators. The same is true for frequency modulation provided the symbol of the operator is piecewise continuous. In this article, it is shown that frequency modulation can destroy Fredholmness for Toeplitz operators with almost periodic symbols; the corresponding example is based on the observation that certain almost periodic functions become semi-almost periodic functions after appropriate frequency modulation. Moreover, this article contains several results that can be employed in order to decide whether a Toeplitz operator with a frequency modulated semi-almost periodic symbol is Fredholm.

Toeplitz Operators with Frequency Modulated Semi-Almost Periodic Symbols

In the theory of traces on operator ideals, it is desirable to treat not only the complex case. Several proofs become much easier when the underlying operators are represented by real matrices. Motivated by this observation, we prove two theorems which, to the best of our knowledge, are not available in the real setting: (1) every operator is a finite linear combination of orthogonal operators, and (2) every skew-symmetric compact operator S is a commutator [A, T], where certain properties of S are inherited to T. In our opinion, theses results are interesting for their own sake. They will also be used in future studies of trace theory by the second-named author.

Albrecht Böttcher

Papers

Toeplitz Operators with PQC Symbols on Weighted Hardy Spaces

The C*-algebra of singular integral operators with semi-almost periodic coefficients

Discrete Wiener-Hopf operators on spaces with Muckenhoupt weight

Toeplitz Operators with Frequency Modulated Semi-Almost Periodic Symbols

Orthogonal and Skew-Symmetric Operators in Real Hilbert Space