Albrecht Böttcher

Analysis of Toeplitz operators

A revised introduction to the advanced analysis of block Toeplitz operators including recent research. This book builds on the success of the first edition which has been used as a standard reference for fifteen years. Topics range from the analysis of locally sectorial matrix functions to Toeplitz and Wiener-Hopf determinants. This will appeal to both graduate students and specialists in the theory of Toeplitz operators.

Spectral Properties of Banded 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.

Carleson curves, Muckenhoupt weights, and Toeplitz operators

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 Cotlars 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 Cotlars 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 Hasemans problem.- 10.10 Notes and comments.

Convolution operators and factorization of almost periodic matrix functions

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.

On the condition numbers of large semidefinite Toeplitz matrices

Abstract This paper is devoted to asymptotic estimates for the (spectral or Euclidean) condition numbers κ ( T n ( a )) = ∥ T n ( a )∥ ∥ T −1 n ( a )∥ of large n × n Toeplitz matrices T n ( a ) in the case where the symbol a is an L ∞ function and Re a ≥ 0 almost everywhere. We describe several classes of symbols a for which κ ( T n ( a )) increases like (log n ) x , n x , or even e xn .

Toeplitz matrices, asymptotic linear algebra, and functional analysis

This text is a self-contained introduction to some problems for Toeplitz matrices that are placed in the borderland between linear algebra and functional analysis. The text looks at Toeplitz matrices with rational symbols, and focuses attention on the asymptotic behavior of the singular values, which includes the behavior of the norms, the norms of the inverses, and the condition numbers as special cases. The text illustrates that the asymptotics of several linear algebra characteristics depend in a fascinating way on functional analytic properties of infinite matrices. Many convergence results can very comfortably be obtained by working with appropriate C*-algebras, while refinements of these results, for example, estimates of the convergence speed, nevertheless require hard analysis.

Toeplitz matrices and determinants with Fisher-Hartwig symbols

Abstract This paper is concerned with Toeplitz matrices generated by symbols of the form a(t) = Φ r=1 R | t−t r | 2α r (− t ) t r β r b(t) (| t |=1) where t 1 ,…, t R are pairwise distinct points on the unit circle, b is sufficiently smooth, b(t) ≠ 0 (¦t¦ = 1) , and ind b = 0, (− t ) t r β r is defined as exp {iβ r arg ( − t t r )} with ¦arg ( − t t r )¦ , and α r , β r are complex numbers satisfying ¦ Re α r ¦ 1 2 , ¦ Re β r ¦ 1 2 . Weighted Hardy spaces L + 2 ( ϱ 1 ) and L + 2 ( ϱ 2 ) are defined such that the Toeplitz operator T ( a ): L + 2 ( ϱ 1 ) → L + 2 ( ϱ 2 ) is bounded and invertible and that the finite section method is applicable to T ( a ) considered as acting from L + 2 ( ϱ 1 ) onto L + 2 ( ϱ 2 ). These results are then applied to prove a conjecture of Fisher and Hartwig ( Adv. in Chem. Phys. 15 (1968), 333–353) which asserts that the determinants of the n × n sections of T ( a ) are asymptotically equal to G(b) n n q E (a) (n → ∞) , where q = ∑ ( α r 2 − β r 2 ) and G ( b ), E (a) are certain nonzero constants. Under quite general conditions on the smoothness of b, these constants are completely identified.

Convergence rates for Tikhonov regularization from different kinds of smoothness conditions

The article is concerned with ill-posed operator equations Ax = y where A:X →Y is an injective bounded linear operator with non-closed range and X and Y are Hilbert spaces. The solution x=x † is assumed to be in the range of some selfadjoint strictly positive bounded linear operator G:X →X. Under several assumptions on G, such as or more generally , inequalities of the form , or range inclusions , convergence rates for the regularization error of Tikhonov regularization are established. We also show that part of our assumptions automatically imply so-called source conditions. The article contains a series of new results but also intends to uncover cross-connections between the different kinds of smoothness conditions that have been discussed in the literature on convergence rates for Tikhonov regularization.

Banach Algebras Generated by N Idempotents and Applications

It is well known that for Banach algebras generated by two idempotents and the identity all irreducible representations are of order not greater than two. These representations have been described completely and have found important applications to symbol theory. It is also well known that without additional restrictions on the idempotents these results do not admit a natural generalization to algebras generated by more than two idempotents and the identity. In this paper we describe all irreducible representations of Banach algebras generated by N idempotents which satisfy some additional relations. These representations are of order not greater than N and allow us to construct a symbol theory with applications to singular integral operators.

Weighted norm inequalities

In this chapter we prove that if Γ is a Carleson curve and w is a weight in A P (Γ) (1 <p < ∞), then the Cauchy singular integral operator S is bounded on L P (Γ,w). There are now various proofs of this deep result, and the proof given in the following is certainly not the most elegant proof. However, it is reasonably self-contained and it contains several details which are usually disposed of as “standard” and are therefore omitted in the advanced texts on this topic.