Toeplitz matrices, asymptotic linear algebra and functional analysis
This small book is a very nice introduction to some problems concerning the approximation of a Toeplitz matrix by its principal submatrices. Recall that for an essentially bounded measurable function \(a\) on the unit circle, the corresponding Toeplitz matrix \(T(a)\) is \([\widehat a_{i-j}] _{i,j=0}^\infty\), where \(\widehat a_j\) are the Fourier coefficients of~\(a\). Let \(T_n(a)=[\widehat a_{i-j}]_{i,j=0}^{n-1}\) be the \(n\times n\) principal submatrix of~\(T(a)\). The authors study the convergence of the norms, the norms of the inverses, the inverses themselves, the spectra, the pseudospectra, and the singular values of \(T_n(a)\) to those of \(T(a)\) as \(n\to\infty\). The basic facts concerning these are established in Chapter~2, after presenting some introductory material (basic properties of Toeplitz operators, basics of \(C^*\)-algebras and Fredholm theory) in Chapter~1. Chapter~3 is concerned with the speed of divergence of \(\|T_n(a)^{-1}\|\) to infinity for non-invertible~\(T(a)\). Chapter~4 discusses the rate of convergence of the norms \(\|T_n(a)\|\), the norms of the inverses \(\|T_n(a)^{-1}\|\), and the condition numbers \(\kappa(T_n(a))=\|T_n(a)\|\|T_n(a)^{-1}\|\) to \(\|T(a)\|\), \(\|T(a)^{-1}\|\) and \(\kappa(T(a))\), respectively, for invertible~\(T(a)\). Finally, in Chapter~5 the limiting behaviour of the singular values is investigated, both the individual limits of \(s_k(T_n(a))\) and \(s_{n-k}(T_n(a))\) as \(k\) is fixed while \(n\to\infty\), and the asymptotic distribution of the singular values (i.e. the limits of the spectra \(\sigma(|T_n(a)|)\) as \(n\) tends to infinity -- the Szegö limit theorem, the Avram-Parter theorem, etc.). The authors usually confine themselves to symbols \(a\) which are rational functions; some information on the general case can be found in \textit{A.~Böttcher} and \textit{B.~Silbermann}'s book ``Introduction to large truncated Toeplitz matrices'', New York (1999; Zbl 0916.15012). The book is fairly self-contained, the reader is required to have only the command of linear algebra and functional analysis up to rudiments of \(C^*\)-algebra theory. In~the reviewer's opinion, this is a good introduction into the subject for beginners, while handsome enough to appeal also to specialists in the area. See the review of the Birkhäuser edition in [Zbl 0969.47022].
scientific article; zbMATH DE number 1448985