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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 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

Preface 1. Infinite matrices 2. Determinants 3. Stability 4. Instability 5. Norms 6. Condition numbers 7. Substitutes for the spectrum 8. Transient behavior 9. Singular values 10. Extreme eigenvalues 11. Eigenvalue distribution 12. Eigenvectors and pseudomodes 13. Structured perturbations 14. Impurities Bibliography Index.

Spectral Properties of Banded Toeplitz Matrices

1 Carleson curves.- 1.1 Definitions and examples.- 1.2 Growth of the argument.- 1.3 Seifullayev bounds.- 1.4 Submultiplicative functions.- 1.5 The W transform.- 1.6 Spirality indices.- 1.7 Notes and comments.- 2 Muckenhoupt weights.- 2.1 Definitions.- 2.2 Power weights.- 2.3 The logarithm of a Muckenhoupt weight.- 2.4 Symmetric and periodic reproduction.- 2.5 Portions versus arcs.- 2.6 The maximal operator.- 2.7 The reverse Holder inequality.- 2.8 Stability of Muckenhoupt weights.- 2.9 Muckenhoupt condition and W transform.- 2.10 Oscillating weights.- 2.11 Notes and comments.- 3 Interaction between curve and weight.- 3.1 Moduli of complex powers.- 3.2 U and V transforms.- 3.3 Muckenhoupt condition and U transform.- 3.4 Indicator set and U transform.- 3.5 Indicator functions.- 3.6 Indices of powerlikeness.- 3.7 Shape of the indicator functions.- 3.8 Indicator functions of prescribed shape.- 3.9 Notes and comments.- 4 Boundedness of the Cauchy singular integral.- 4.1 The Cauchy singular integral.- 4.2 Necessary conditions for boundedness.- 4.3 Special curves and weights.- 4.4 Brief survey of results on general curves and weights.- 4.5 Composing curves and weights.- 4.6 Notes and comments.- 5 Weighted norm inequalities.- 5.1 Again the maximal operator.- 5.2 The Calderon-Zygmund decomposition.- 5.3 Cotlar's inequality.- 5.4 Good ? inequalities.- 5.5 Modified maximal operators.- 5.6 The maximal singular integral operator.- 5.7 Lipschitz curves.- 5.8 Measures in the plane.- 5.9 Cotlar's inequality in the plane.- 5.10 Maximal singular integrals in the plane.- 5.11 Approximation by Lipschitz curves.- 5.12 Completing the puzzle.- 5.13 Notes and comments.- 6 General properties of Toeplitz operators.- 6.1 Smirnov classes.- 6.2 Weighted Hardy spaces.- 6.3 Fredholm operators.- 6.4 Toeplitz operators.- 6.5 Adjoints.- 6.6 Two basic theorems.- 6.7 Hankel operators.- 6.8 Continuous symbols.- 6.9 Classical Toeplitz matrices.- 6.10 Separation of discontinuities.- 6.11 Localization.- 6.12 Wiener-Hopf factorization.- 6.13 Notes and comments.- 7 Piecewise continuous symbols.- 7.1 Local representatives.- 7.2 Fredholm criterion.- 7.3 Leaves and essential spectrum.- 7.4 Metamorphosis of leaves.- 7.5 Logarithmic leaves.- 7.6 General leaves.- 7.7 Index and spectrum.- 7.8 Semi-Fredholmness.- 7.9 Notes and comments.- 8 Banach algebras.- 8.1 General theorems.- 8.2 Operators of local type.- 8.3 Algebras generated by idempotents.- 8.4 An N projections theorem.- 8.5 Algebras associated with Jordan curves.- 8.6 Notes and comments.- 9 Composed curves.- 9.1 Extending Carleson stars.- 9.2 Extending Muckenhoupt weights.- 9.3 Operators on flowers.- 9.4 Local algebras.- 9.5 Symbol calculus.- 9.6 Essential spectrum of the Cauchy singular integral.- 9.7 Notes and comments.- 10 Further results.- 10.1 Matrix case.- 10.2 Index formulas.- 10.3 Kernel and cokernel dimensions.- 10.4 Spectrum of the Cauchy singular integral.- 10.5 Orlicz spaces.- 10.6 Mellin techniques.- 10.7 Wiener-Hopf integral operators.- 10.8 Zero-order pseudodifferential operators.- 10.9 Conformal welding and Haseman's problem.- 10.10 Notes and comments.

Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators

1 Convolution Operators and Their Symbols.- 2 Introduction to Scalar Wiener-Hopf Operators.- 3 Scalar Wiener-Hopf Operators with SAP Symbols.- 4 Some Phenomena Caused by SAP Symbols.- 5 Introduction to Matrix Wiener-Hopf Operators.- 6 Factorization of Matrix Functions.- 7 Bohr Compactification.- 8 Existence and Uniqueness ofAPFactorization.- 9 Matrix Wiener-Hopf Operators withAPWSymbols.- 10 Matrix Wiener-Hopf Operators withSAPWSymbols.- 11 Left Versus Right Wiener-Hopf Factorization.- 12 Corona Theorems.- 13 The Portuguese Transformation.- 14 Some Concrete Factorizations.- 15 Scalar Trinomials.- 16 Toeplitz Operators.- 17 Zero-Order Pseudodifferential Operators.- 18 Toeplitz Operators with SAP Symbols on Hardy Spaces.- 19 Wiener-Hopf Operators with SAP Symbols on Lebesgue Spaces.- 20 Hankel Operators on Besicovitch Spaces.- 21 Generalized AP Factorization.- 22 Canonical Wiener-Hopf Factorization via Corona Problems.- 23 Canonical APW Factorization via Corona Problems.

Albrecht Böttcher

Papers

Analysis of Toeplitz Operators

Introduction to Large Truncated Toeplitz Matrices

Spectral Properties of Banded Toeplitz Matrices

Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators

Convolution Operators and Factorization of Almost Periodic Matrix Functions