# Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics

This book is a volume 259 in the Bikhäuser OT Series *Operator Theory Advances and Applications*. It contains 30 contributions celebrating Albert Böttcher's 60th birthday.

Albert Böttcher is a professor of mathematics at the TU Chemnitz in Germany. His main research topic is functional analysis. At his 18th he won the silver medal at the International Math Olympiad in Moscow. He studied mathematics at the TU Karl-Marx-Stadt (now TU Chemnitz) and finished a PhD in 1984 entitled *The finite section method for the Wiener-Hopf integral operator* under supervision of V.B. Dybin at Rostov-on-Don State University in Russia (this was all while the Berlin wall was still up). Since then he stayed at the TU Chemnitz. At the time of writing he has (co)authored 9 books and over 220 papers. The complete list is in the beginning of the book but one may also consult his website where he keeps his list of publications up to date.

The contributions start with reminiscences and best wishes by friends, colleagues and students of Albrecht Böttcher. Besides personal recollections, there is some discussion of his work, some photographs and reproductions of slides he used in presentations to illustrate that he is not only an excellent mathematician but also a passionate teacher and lecturer.

That leaves about 700 pages of original research papers all of which relate from far or near to subjects that Böttcher has worked on. The Toeplitz operators and Toeplitz matrices of the title are indeed well represented, but there are all the other "Related Topics" which are close to his work too. About fifty renowned authors are involved.

The Toeplitz operator (and hence also its spectrum) is characterized by a function, which is called its symbol. It features in a multiplication or convolution in the definition of the operator. With respect to a standard monomial basis, Toeplitz operators are represented by (infinite) Toeplitz matrices that have constant entries along diagonals. Of course the spectral and other properties of truncations of the infinite matrices to large finite ones relate to corresponding properties of Toeplitz operators, and similarly it can be related to other operators such as convolution and Wiener-Hopf operators. These matrices and operators have applications in differential and integral equations, systems and control, signal processing, and many more. Depending on the application the symbol may get an interpretation of transfer function of a system, power spectrum or autocorrelation of a signal, the kernel of an integral equation, or just a weight function in a Hilbert space. So, Toeplitz matrices and operators are also related to numerical methods for solving functional equations after discretization. Or to orthogonal polynomials (on the unit circle), which then in turn links to (trigonometric) moment problems, quadrature, and approximation theory (on the unit circle, but in a similar way also to analogs on the real line).

Obviously this is not the place to discuss every paper in detail. The table of contents is available on the publisher's website and for convenience the research papers are also listed below. From the titles you will recognize the papers on determinants and eigenvalues for Toeplitz matrices, in particular their asymptotic behaviour as their size goes to infinity. Of course circulant and Hankel operators and combinations of these as operators or matrices are not far off the central theme and they are thus also treated in some of the chapters. The majority of the papers present new results. Note that most of them are (functional) analysis. Only a few exceptions are more linear algebra or make a link to physics or explicitly discuss numerical aspects (see [14, 16, 18, 23, 25, 27] below).

Some of the papers are quite long (more than 30 pages and some even up to 50 pages). They are basically true research papers, sometimes a bit more expository, but they are not of the introductory broad survey type. So this is not the book you should read to be introduced to the subject, but is is more a sketch of the state-of-the-art for who is already famiiar. The style of course depends on the authors, but the book is homogeneous because of the subjects that all somehow relate to Böttcher's work. These topics discussed here are also close to the core idea of this book series *Operators Theory Advances and Applications*, founded by Israel Gohberg as a complement to the journal *Integral Equations and Operator Theory*. Only one of Böttcher's books appeared in this series though (*Convolution Operators and Factorization of Almost Periodic Matrix Functions * (2002) authored with Yu. I. Karlovich, and I. M. Spitkovsky appeared as volume 131) but several of his books are with Springer / Birkhäuser. That these topics are still a main focus of research is illustrated by the successful annual IWOTA conferences (*International Workshop on Operator Theory and its Applications*), the proceedings of which are also published in this OT series. The IWOTA 2017 is organized by A. Böttcher, D. Potts and P. Stollmann at the TU Chemnitz.

Thus for anyone interested in the general topics of this book series, this collection will be a worthy addition. For those who are more selective, there is of course still the possibility to get some separate chapters, which is the advantage of having it also available as an ebook.

Here are the titles and authors of the research papers in this volume:

7. *Asymptotics of Eigenvalues for Pentadiagonal Symmetric Toeplitz Matrices, * Barrera, M. (et al.), Pages 51-77

8. *Echelon Type Canonical Forms in Upper Triangular Matrix Algebras, * Bart, H. (et al.), Pages 79-124

9. *Asymptotic Formulas for Determinants of a Special Class of Toeplitz + Hankel Matrices, * Basor, E. (et al.), Pages 125-154

10. *Generalization of the Brauer Theorem to Matrix Polynomials and Matrix Laurent Series, * Bini, D.A. (et al.), Pages 155-178

11. *Eigenvalues of Hermitian Toeplitz Matrices Generated by Simple-loop Symbols with Relaxed Smoothness, * Bogoya, J.M. (et al.), Pages 179-212

12. *On the Asymptotic Behavior of a Log Gas in the Bulk Scaling Limit in the Presence of a Varying External Potential II, * Bothner, T. (et al.), Pages 213-234

13. *Useful Bounds on the Extreme Eigenvalues and Vectors of Matrices for Harper's Operators, * Bump, D. (et al.), Pages 235-265

14. *Fast Inversion of Centrosymmetric Toeplitz-plus-Hankel Bezoutians, * Ehrhardt, T. (et al.), Pages 267-300

15. *On Matrix-valued Stieltjes Functions with an Emphasis on Particular Subclasses, * Fritzsche, B. (et al.), Pages 301-352

16. *The Theory of Generalized Locally Toeplitz Sequences: a Review, an Extension, and a Few Representative Applications, * Garoni, C. (et al.), Pages 353-394

17. *The Bézout Equation on the Right Half-plane in a Wiener Space Setting, * Groenewald, G.J. (et al.), Pages 395-411

18. *On a Collocation-quadrature Method for the Singular Integral Equation of the Notched Half-plane Problem, * Junghanns, P. (et al.), Pages 413-462

19. *The Haseman Boundary Value Problem with Slowly Oscillating Coefficients and Shifts, * Karlovich, Yu.I., Pages 463-500

20. *On the Norm of Linear Combinations of Projections and Some Characterizations of Hilbert Spaces, * Krupnik, N. (et al.), Pages 501-510

21. *Pseudodifferential Operators in Weighted Hölder-Zygmund Spaces of Variable Smoothness, * Kryakvin, V. (et al.), Pages 511-531

22. *Commutator Estimates Comprising the Frobenius Norm - Looking Back and Forth, * Lu, Zhiqin (et al.), Pages 533-559

23. *Numerical Ranges of 4-by-4 Nilpotent Matrices: Flat Portions on the Boundary, * Militzer, E. (et al.), Pages 561-591

24. *Traces on Operator Ideals and Related Linear Forms on Sequence Ideals (Part IV), * Pietsch, A., Pages 593-619

25. *Error Estimates for the ESPRIT Algorithm, * Potts, D. (et al.), Pages 621-648

26. *The Universal Algebra Generated by a Power Partial Isometry, * Roch, S., Pages 649-662

27. *Norms, Condition Numbers and Pseudospectra of Convolution Type Operators on Intervals, * Seidel, M., Pages 663-680

28. *Paired Operators in Asymmetric Space Setting, * Speck, F.-O., Pages 681-702

29. *Natural Boundary for a Sum Involving Toeplitz Determinants, * Tracy, C.A. (et al.), Pages 703-718

30. *A Riemann-Hilbert Approach to Filter Design, * Wegert, E., Pages 719-740

**Submitted by Adhemar Bultheel |

**20 / May / 2017