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Introduction to large truncated Toeplitz matrices. (English) Zbl 0916.15012
Universitext. New York, NY: Springer. xi, 258 p. (1999).
The aim of the book is the investigation of large finite Toeplitz matrices. Three main topics are discussed: pseudospectra, singular values, eigenvalues. Large finite Toeplitz matrices are considered as truncations of infinite Toeplitz matrices and properties of an individual large finite Toeplitz matrix follow from properties of a sequence of the truncations (finite sections) of an infinite Toeplitz matrix. The investigation of characteristics of truncated Toeplitz matrices is based on the idea of stability. Given an infinite Toeplitz matrix $$A$$, let $$\{A_{n}\}_{n=1}^{\infty }$$stand for the sequence $$n\times n$$ truncations. The sequence $$\{A_{n}\}_{n=1}^{\infty}$$ is called stable if there exists $$n_0$$ such that the matrices $$A_{n}$$ are invertible for $$n\geq n_{0}$$ and $$\sup_{n\geq n_{0}}\| A_{n}^{-1}\| <\infty$$. The investigation of stability is given in the book as a nice application of the theory of $$C^{\ast }-$$algebras and local principles.
The book, written by the well-known specialists in the Toeplitz and singular integral operators and numerical methods for them, is a beautiful introduction to the theory of Toeplitz operators, finite sections methods, limit properties of the important characteristics of the truncated Toeplitz matrices such as the condition numbers, Moore-Penrose inverses, traces and determinants. The book contains the full list of bibliographical references and many nice figures, illustrating the theoretical material. The book is an excellent guide for beginners and amateurs. Moreover it is very interesting for the specialists in the Toeplitz and singular integral operators and numerical methods for them.

##### MSC:
 15B05 Toeplitz, Cauchy, and related matrices 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators 47-02 Research exposition (monographs, survey articles) pertaining to operator theory 15-02 Research exposition (monographs, survey articles) pertaining to linear algebra 15A18 Eigenvalues, singular values, and eigenvectors