The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

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

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

We study polynomial interpolation on a d-dimensional cube, where d is large. We suggest to use the least solution at sparse grids with the extrema of the Chebyshev polynomials. The polynomial exactness of this method is almost optimal. Our error bounds show that the method is universal, i.e., almost optimal for many different function spaces. We report on numerical experiments for d = 10 using up to 652 065 interpolation points.

High dimensional polynomial interpolation on sparse grids

First published in 1975, this classic book gives a systematic account of transcendental number theory, that is those numbers which cannot be expressed as the roots of algebraic equations having rational coefficients. Their study has developed into a fertile and extensive theory enriching many branches of pure mathematics. Expositions are presented of theories relating to linear forms in the logarithms of algebraic numbers, of Schmidt's generalisation of the Thue-Siegel-Roth theorem, of Shidlovsky's work on Siegel's |E|-functions and of Sprindzuk's solution to the Mahler conjecture. The volume was revised in 1979: however Professor Baker has taken this further opportunity to update the book including new advances in the theory and many new references.

Transcendental number theory, by Alan Baker. Pp. x, 147. £4·90. 1975. SBN 0 521 20461 5 (Cambridge University Press)

Toeplitz operators arise in plenty of applications. They constitute one of the most important classes of non-selfadjoint operators, and the ideas and methods prevailing in the field of Toeplitz operators are a fascinating illustration of the fruitful interplay between operator theory, complex analysis, and Banach algebra techniques. This book is a systematic introduction to the advanced analysis of block Toeplitz operators and includes both classical results and recent developments.

Its first edition has been a standard reference for fifteen years.

The present second edition is enriched by several results obtained only in the last decade. The topics treated range from the analysis of locally sectorial matrix functions through Toeplitz and Wiener-Hopf operators on Banach spaces, projection methods, and quarter-plane operators up to Toeplitz and Wiener-Hopf determinants.

The book is addressed to both graduate students approaching the subject for the first time and specialists in the theory of Toeplitz operators, but should also be of interest to physicists, probabilists, and computer scientists.

Analysis of Toeplitz Operators

1 Infinite Matrices.- 1.1 Boundedness and Invertibility.- 1.2 Laurent Matrices.- 1.3 Toeplitz Matrices.- 1.4 Hankel Matrices.- 1.5 Wiener-Hopf Factorization.- 1.6 Continuous Symbols.- 1.7 Locally Sectorial Symbols.- 1.8 Discontinuous Symbols.- 2 Finite Section Method and Stability.- 2.1 Approximation Methods.- 2.2 Continuous Symbols.- 2.3 Asymptotic Inverses.- 2.4 The Gohberg-Feldman Approach.- 2.5 Algebraization of Stability.- 2.6 Local Principles.- 2.7 Localization of Stability.- 3 Norms of Inverses and Pseudospectra.- 3.1C*-Algebras.- 3.2 Continuous Symbols.- 3.3 Piecewise Continuous Symbols.- 3.4 Norm of the Resolvent.- 3.5 Limits of Pseudospectra.- 3.6 Pseudospectra of Infinite Toeplitz Matrices.- 4 Moore-Penrose Inverses and Singular Values.- 4.1 Singular Values of Matrices.- 4.2 The Lowest Singular Value.- 4.3 The Splitting Phenomenon.- 4.4 Upper Singular Values.- 4.5 Moler's Phenomenon.- 4.6 Limiting Sets of Singular Values.- 4.7 The Moore-Penrose Inverse.- 4.8 Asymptotic Moore-Penrose Inversion.- 4.9 Moore-Penrose Sequences.- 4.10 Exact Moore-Penrose Sequences.- 4.11 Regularization and Kato Numbers.- 5 Determinants and Eigenvalues.- 5.1 The Strong Szegoe Limit Theorem.- 5.2 Ising Model and Onsager Formula.- 5.3 Second-Order Trace Formulas.- 5.4 The First Szegoe Limit Theorem.- 5.5 Hermitian Toeplitz Matrices.- 5.6 The Avram-Parter Theorem.- 5.7 The Algebraic Approach to Trace Formulas.- 5.8 Toeplitz Band Matrices.- 5.9 Rational Symbols.- 5.10 Continuous Symbols.- 5.11 Fisher-Hartwig Determinants.- 5.12 Piecewise Continuous Symbols.- 6 Block Toeplitz Matrices.- 6.1 Infinite Matrices.- 6.2 Finite Section Method and Stability.- 6.3 Norms of Inverses and Pseudospectra.- 6.4 Distribution of Singular Values.- 6.5 Asymptotic Moore-Penrose Inversion.- 6.6 Trace Formulas.- 6.7 The Szegoe-Widom Limit Theorem.- 6.8 Rational Matrix Symbols.- 6.9 Multilevel Toeplitz Matrices.- 7 Banach Space Phenomena.- 7.1 Boundedness.- 7.2 Fredholmness and Invertibility.- 7.3 Continuous Symbols.- 7.4 Piecewise Continuous Symbols.- 7.5 Loss of Symmetry.- References.- Symbol Index.

Introduction to Large Truncated Toeplitz Matrices

Numerical Analysis for Integral and Related Operator Equations

1 Limit Operators.- 1.1 Generalized compactness, generalized convergence.- 1.1.1 Compactness, strong convergence, Fredholmness.- 1.1.2 P -compactness.- 1.1.3 P -Fredholmness.- 1.1.4 P -strong convergence.- 1.1.5 Invertibility of P -strong limits.- 1.2 Limit operators.- 1.2.1 Limit operators and the operator spectrum.- 1.2.2 Operators with rich operator spectrum.- 1.3 Algebraization.- 1.3.1 Algebraization by restriction.- 1.3.2 Symbol calculus.- 1.4 Comments and references.- 2 Fredholmness of Band-dominated Operators.- 2.1 Band-dominated operators.- 2.1.1 Function spaces on $${\mathbb{Z}^N}$$.- 2.1.2 Matrix representation.- 2.1.3 Operators of multiplication.- 2.1.4 Band and band-dominated operators.- 2.1.5 Limit operators of band-dominated operators.- 2.1.6 Rich band-dominated operators.- 2.2 P-Fredholmness of rich band-dominated operators.- 2.2.1 The main theorem on P-Fredholmness.- 2.2.2 Weakly sufficient families of homomorphisms.- 2.2.3 Symbol calculus for rich band-dominated operators.- 2.2.4 Appendix A: Second version of a symbol calculus.- 2.2.5 Appendix B: Commutative Banach algebras.- 2.3 Local P-Fredholmness: elementary theory.- 2.3.1 Local operator spectra and local invertibility.- 2.3.2 PR-compactness, PR -Fredholmness.- 2.3.3 Local P-Fredholmness of band-dominated operators.- 2.3.4 Allan's local principle.- 2.3.5 Local P-Fredholmness of band-dominated operators in the sense of the local principle.- 2.3.6 Operators with continuous coefficients.- 2.4 Local P-Fredholmness: advanced theory.- 2.4.1 Slowly oscillating functions.- 2.4.2 The maximal ideal space of $$SO\left( { {\mathbb{Z}^N}} \right)$$.- 2.4.3 Preliminaries on nets.- 2.4.4 Limit operators with respect to nets.- 2.4.5 Local invertibility at points in $${M^\infty }\left( {SO\left( { {\mathbb{Z}^N}} \right)} \right)$$.- 2.4.6 Fredholmness of band-dominated operators with slowly oscillating coefficients.- 2.4.7 Nets vs. sequences.- 2.4.8 Appendix A: A second proof of Theorem 2 4 27.- 2.4.9 Appendix B: A third proof of Theorem 2 4 27.- 2.5 Operators in the discrete Wiener algebra.- 2.5.1 The Wiener algebra.- 2.5.2 Fredholmness of operators in the Wiener algebra.- 2.6 Band-dominated operators with special coefficients.- 2.6.1 Band-dominated operators with almost periodic coefficients.- 2.6.2 Operators on half-spaces.- 2.6.3 Operators on polyhedral convex cones.- 2.6.4 Composed band-dominated operators on $${\mathbb{Z}^2}$$.- 2.6.5 Difference operators of second order.- 2.6.6 Discrete Schroedinger operators.- 2.7 Indices of Fredholm band-dominated operators.- 2.7.1 Main results.- 2.7.2 The algebra $$\mathcal{A}\left( \mathbb{Z} \right)$$ as a crossed product.- 2.7.3 The Kl-group of $$\mathcal{A}\left( \mathbb{Z} \right)$$.- 2.7.4 The Kl-group of A+/-.- 2.7.5 Proof of Theorem 2.7.1.- 2.7.6 Unitary band-dominated operators.- 2.8 Comments and references.- 3 Convolution Type Operators on $${\mathbb{R}^N}$$.- 3.1 Band-dominated operators on $${L^p}\left( { {\mathbb{R}^N}} \right)$$.- 3.1.1 Approximate identities and P-Fredholmness.- 3.1.2 Shifts and limit operators.- 3.1.3 Discretization.- 3.1.4 Band-dominated operators on $${L^p}\left( { {\mathbb{R}^N}} \right)$$.- 3.2 Operators of convolution.- 3.2.1 Compactness of semi-commutators.- 3.2.2 Compactness of commutators.- 3.3 Fredholmness of convolution type operators.- 3.3.1 Operators of convolution type.- 3.3.2 Fredholmness.- 3.4 Compressions of convolution type operators.- 3.4.1 Compressions of operators in $$\mathcal{A}\left( {BUC\left( { {\mathbb{R}^N}} \right),{\mathcal{C}_p}} \right)$$.- 3.4.2 Compressions to a half-space.- 3.4.3 Compressions to curved half-spaces.- 3.4.4 Compressions to curved layers.- 3.4.5 Compressions to curved cylinders.- 3.4.6 Compressions to cones with smooth cross section.- 3.4.7 Compressions to cones with edges.- 3.4.8 Compressions to epigraphs of functions.- 3.5 A Wiener algebra of convolution-type operators.- 3.5.1 Fredholmness of operators in the Wiener algebra.- 3.5.2 The essential spectrum of Schroedinger operators.- 3.6 Comments and references.- 4 Pseudodifferential Operators.- 4.1 Generalities and notation.- 4.1.1 Function spaces and Fourier transform.- 4.1.2 Oscillatory integrals.- 4.1.3 Pseudodifferential operators.- 4.1.4 Formal symbols.- 4.1.5 Pseudodifferential operators with double symbols.- 4.1.6 Boundedness on $${L^2}\left( { {\mathbb{R}^N}} \right)$$.- 4.1.7 Consequences of the Calderon-Vaillancourt theorem.- 4.2 Bi-discretization of operators on $${L^2}\left( { {\mathbb{R}^N}} \right)$$.- 4.2.1 Bi-discretization.- 4.2.2 Bi-discretization and Fredholmness.- 4.2.3 Bi-discretization and limit operators.- 4.3 Fredholmness of pseudodifferential operators.- 4.3.1 A Wiener algebra on $${L^2}\left( { {\mathbb{R}^N}} \right)$$.- 4.3.2 Fredholmness of operators in $${\mathcal{W}^\$ }\left( { {L^2}\left( { {\mathbb{R}^N}} \right)} \right)$$.- 4.3.3 Fredholm properties of pseudodifferential operators in OPS0,00.- 4.4 Applications.- 4.4.1 Operators with slowly oscillating symbols.- 4.4.2 Operators with almost periodic symbols.- 4.4.3 Operators with semi-almost periodic symbols.- 4.4.4 Pseudodifferential operators of nonzero order.- 4.4.5 Differential operators.- 4.4.6 Schroedinger operators.- 4.4.7 Partial differential-difference operators.- 4.5 Mellin pseudodifferential operators.- 4.5.1 Pseudodifferential operators with analytic symbols.- 4.5.2 Mellin pseudodifferential operators.- 4.5.3 Mellin pseudodifferential operators with analytic symbols.- 4.5.4 Local invertibility of Mellin pseudodifferential operators.- 4.6 Singular integrals over Carleson curves with Muckenhoupt weights.- 4.6.1 Carleson curves and Muckenhoupt weights.- 4.6.2 Logarithmic spirals and power weights.- 4.6.3 Curves and weights with slowly oscillating data.- 4.6.4 Local representatives and local spectra of singular integral operators.- 4.6.5 Singular integral operators on composed curves.- 4.7 Comments and references.- 5 Pseudodifference Operators.- 5.1 Pseudodifference operators.- 5.2 Fredholmness of pseudodifference operators.- 5.3 Fredholm properties of pseudodifference operators on weighted spaces.- 5.3.1 Boundedness on weighted spaces.- 5.3.2 Fredholmness on weighted spaces.- 5.3.3 The Phragmen-Lindeloef principle.- 5.4 Slowly oscillating pseudodifference operators.- 5.4.1 Fredholmness on lP-spaces.- 5.4.2 Fredholmness on weighted spaces, Phragmen-Lindeloef principle.- 5.4.3 Fredholm index for operators in OPSO.- 5.5 Almost periodic pseudodifference operators.- 5.6 Periodic pseudodifference operators.- 5.6.1 The one-dimensional case.- 5.6.2 The multi-dimensional case.- 5.7 Semi-periodic pseudodifference operators.- 5.7.1 Fredholmness on unweighted spaces.- 5.7.2 Fredholmness on weighted spaces.- 5.7.3 Fredholm index.- 5.8 Discrete Schroedinger operators.- 5.8.1 Slowly oscillating potentials.- 5.8.2 Exponential decay of eigenfunctions.- 5.8.3 Semi-periodic Schroedinger operators.- 5.9 Comments and references.- 6 Finite Sections of Band-dominated Operators.- 6.1 Stability of the finite section method.- 6.1.1 Approximation sequences.- 6.1.2 Stability vs. invertibility.- 6.1.3 Stability vs. Fredholmness.- 6.2 Finite sections of band-dominated operators on $${\mathbb{Z}^1}$$ and $${\mathbb{Z}^2}$$.- 6.2.1 Band-dominated operators on $${\mathbb{Z}^1}$$: the general case.- 6.2.2 Band-dominated operators on $${\mathbb{Z}^1}$$: slowly oscillating coefficients.- 6.2.3 Band-dominated operators on $${\mathbb{Z}^2}$$.- 6.2.4 Finite sections of convolution type operators.- 6.3 Spectral approximation.- 6.3.1 Weakly sufficient families and spectra.- 6.3.2 Interlude: Spectra of band-dominated operators on Hilbert spaces.- 6.3.3 Asymptotic behavior of norms.- 6.3.4 Asymptotic behavior of spectra.- 6.4 Fractality of approximation methods.- 6.4.1 Fractal approximation sequences.- 6.4.2 Fractality and norms.- 6.4.3 Fractality and spectra.- 6.4.4 Fractality of the finite section method for a class of band-dominated operators.- 6.5 Comments and references.- 7 Axiomatization of the Limit Operators Approach.- 7.1 An axiomatic approach to the limit operators method.- 7.2 Operators on homogeneous groups.- 7.2.1 Homogeneous groups.- 7.2.2 Multiplication operators.- 7.2.3 Partition of unity.- 7.2.4 Convolution operators.- 7.2.5 Shift operators.- 7.3 Fredholm criteria for convolution type operators with shift.- 7.3.1 Operators on homogeneous groups.- 7.3.2 Operators on discrete subgroups.- 7.4 Comments and references.

Limit operators and their applications in operator theory

The language of numerical analysis regularization approximation of spectra stability analysis representation theory Fredholm sequences self-adjoint approximation sequences.

Bernd Silbermann

Papers

Analysis of Toeplitz Operators

Introduction to Large Truncated Toeplitz Matrices

Numerical Analysis for Integral and Related Operator Equations

Limit operators and their applications in operator theory

C* - Algebras and Numerical Analysis