Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Acknowledgements
- Other
- Figures
- 1 Why Toeplitz–Hankel? Motivations and Panorama
- 2 Hankel and Toeplitz: Sibling Operators on the Space H2
- 3 H2 Theory of Toeplitz Operators
- 4 Applications: Riemann–Hilbert, Wiener–Hopf, Singular Integral Operators (SIO)
- 5 Toeplitz Matrices: Moments, Spectra, Asymptotics
- Appendix A Key Notions of Banach Spaces
- Appendix B Key Notions of Hilbert Spaces
- Appendix C An Overview of Banach Algebras
- Appendix D Linear Operators
- Appendix E Fredholm Operators and the Noether Index
- Appendix F A Brief Overview of Hardy Spaces
- References
- Backmatter
- Index
5 - Toeplitz Matrices: Moments, Spectra, Asymptotics
Published online by Cambridge University Press: 06 December 2019
- Frontmatter
- Dedication
- Contents
- Preface
- Acknowledgements
- Other
- Figures
- 1 Why Toeplitz–Hankel? Motivations and Panorama
- 2 Hankel and Toeplitz: Sibling Operators on the Space H2
- 3 H2 Theory of Toeplitz Operators
- 4 Applications: Riemann–Hilbert, Wiener–Hopf, Singular Integral Operators (SIO)
- 5 Toeplitz Matrices: Moments, Spectra, Asymptotics
- Appendix A Key Notions of Banach Spaces
- Appendix B Key Notions of Hilbert Spaces
- Appendix C An Overview of Banach Algebras
- Appendix D Linear Operators
- Appendix E Fredholm Operators and the Noether Index
- Appendix F A Brief Overview of Hardy Spaces
- References
- Backmatter
- Index
Summary
Trigonometric moment problem, positive definite Toeplitz matrices, orthogonal polynomials.
Arveson’s distance formula, the four-block lemma, Julia’s lemma.
Inversion of a Toeplitz matrix by Gohberg-Sementsul and Baxter-Hirschman, inversion by the finite section method.
Circulants.
Toeplitz determinants and their asymptotics, equidistribution following Weyl, asymptotic distribution of Toeplitz spectra (following Szegő) and of circulants.
The “strong Szegő theorem,” Krein-Dirichlet algebra.
The Helton-Howe and Berger-Shaw formulas of det/trace.
Positive harmonic functions, holomorphic functions of the Schur class
Inversion of Wiener–Hopf operators by the finite section method, case of a rational symbol.
Cauchy and Hilbert determinants, Libkind-Widom and Borodin Okounkov formulas.
- Type
- Chapter
- Information
- Toeplitz Matrices and Operators , pp. 230 - 328Publisher: Cambridge University PressPrint publication year: 2020