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Introduction to Large Truncated Toeplitz Matrices is a text on the application of functional analysis and operator theory to some concrete asymptotic problems of linear algebra. The book contains results on the stability of projection methods, deals with asymptotic inverses and Moore-Penrose inversion of large Toeplitz matrices, and embarks on the asymptotic behavoir of the norms of inverses, the pseudospectra, the singular values, and the eigenvalues of large Toeplitz matrices. The approach is heavily based on Banach algebra techniques and nicely demonstrates the usefulness of C*-algebras and local principles in numerical analysis. The book includes classical topics as well as results obtained and methods developed only in the last few years. Though employing modern tools, the exposition is elementary and aims at pointing out the mathematical background behind some interesting phenomena one encounters when working with large Toeplitz matrices. The text is accessible to readers with basic knowledge in functional analysis. It is addressed to graduate students, teachers, and researchers with some inclination to concrete operator theory and should be of interest to everyone who has to deal with infinite matrices (Toeplitz or not) and their large truncations.



1. Infinite Matrices

The purpose of this section is to fix some standard notations and to recall some terminology.
Bernd Silbermann

2. Finite Section Method and Stability

Let \( A = \left( {{a_{jk}}} \right)_{j,k = 1}^\infty \) be an infinite matrix and suppose A generates a bounded operator on l2. In order to solve the equation Ax = y, i.e., the infinite linear system
$$ \left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{a_{11}}}{{a_{12}}}{{a_{13}}} \ldots \end{array}} \\ {\begin{array}{*{20}{c}} {{a_{21}}}{{a_{22}}}{{a_{23}}} \ldots \end{array}} \\ {\begin{array}{*{20}{c}} {{a_{31}}}{{a_{32}}}{{a_{33}}} \ldots \end{array}} \\ {\begin{array}{*{20}{c}} \ldots \ldots \ldots \ldots \end{array}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{x_1}} \\ {{x_2}} \\ {{x_3}} \\ \vdots \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {{y_1}} \\ {{y_2}} \\ {{y_3}} \\ \vdots \end{array}} \right) $$
we consider the truncated systems
$$ \left( {\begin{array}{*{20}c} {a_{11} } & \ldots & {a_{1n} } \\ \vdots & {} & \vdots \\ {a_{n1} } & \cdots & {a_{nn} } \\ \end{array}} \right)\left( \begin{array}{l} x_1^{(n)} \\ \vdots \\ x_n^{(n)} \\ \end{array} \right) = \left( \begin{array}{l} y_1 \\ \vdots \\ y_n \\ \end{array} \right). $$
Albrecht Böttcher, Bernd Silbermann

3. Norms of Inverses and Pseudospectra

C*-algebras are especially nice Banach algebras. A map \( a \mapsto {a^*} \) of a Banach algebra A onto itself is called an involution if
$$ {a^{**}} = a,{\left( {a + b} \right)^*},{\left( {ab} \right)^*} = {b^*}{a^*},{\left( {\lambda a} \right)^*} = \mathop \lambda \limits^ -{a^*} $$
for all a, b ∈ A and all.λ ∈ C. A C*-algebra is a Banach algebra with an involution such that
$$ \left\| {aa^* } \right\| = \left\| a \right\|^2 {\rm{ for all }}a \in A. $$
Albrecht Böttcher, Bernd Silbermann

4. Moore-Penrose Inverses and Singular Values

Let H be a Hilbert space and let A be a bounded linear operator on H. Then sp A*A ⊂[0,∞), and the non-negative square roots of the numbers in sp A*A are called the singular values of A. The set of the singular values of A will be denoted byΣ(A),
$$ \sum {\left( A \right)} : = \left\{ {s \in \left[ {\left. {0,\infty } \right):{s^2} \in sp{A^*}A} \right.} \right\}. $$
Albrecht Böttcher, Bernd Silbermann

5. Determinants and Eigenvalues

We now study the behavior of the determinants
$${D_n}(a): = \det {T_n}(a): = \det \left( {\begin{array}{*{20}{c}} {{a_0}}&{{a_{ - 1}}}& \cdots &{{a_{ - (n - 1)}}} \\ {{a_1}}&{{a_0}}& \cdots &{{a_{ - (n - 2)}}} \\ \vdots & \vdots & \ddots & \vdots \\ {{a_{n - 1}}}&{{a_{n - 2}}}& \cdots &{{a_0}} \end{array}} \right)$$
as n goes to infinity. The strong Szegö limit theorem says that, after appropriate normalization, the determinants Dn(a) approach a nonzero limit provided a is sufficiently smooth and T(a) is invertible. Before stating and proving this theorem, we need a few more auxiliary facts.
Albrecht Böttcher, Bernd Silbermann

6. Block Toeplitz Matrices

A Toeplitz matrix is constant along the parallels to the main diagonal. Matrices whose entries in the parallels to the main diagonal form periodic sequences (with the same period N) are referred to as block Toeplitz matrices. Equivalently, A is a block Toeplitz matrix if and only if
$$ A = \left( {\begin{array}{*{20}{c}} {{a_0}}{{a_{ - 1}}}{{a_{ - 2}}} \cdots \\ {{a_1}}{{a_0}}{{a_{ - 1}}} \cdots \\ {{a_2}}{{a_1}}{{a_0}} \cdots \\ \cdots \cdots \cdots \cdots \end{array}} \right) $$
where \(\{ a_k \} _{k \in z} \) is a sequence of N × N matrices,\( {a_k} \in B({C^N}) \) for all k ∈ Z.
Albrecht Böttcher, Bernd Silbermann

7. Banach Space Phenomena

For 1≤p<∞,µ∈R we denote by \(l_\mu ^p\left( J \right)\) the Banach space of all sequences x=|xj}j∈J such that
$$ {\left\| x \right\|_{p,\mu }}: = {\left( {{{\sum\limits_{j \in J} {{{\left( {\left| j \right| + 1} \right)}^{p\mu }}\left| {{x_j}} \right|} }^p}} \right)^{1/p}}<\infty $$
Albrecht Böttcher, Bernd Silbermann


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