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**Operators, functions, and systems: an easy reading. Volume I: Hardy, Hankel, and Toeplitz. Transl. from the French by Andreas Hartmann.**
*(English)*
Zbl 1007.47001

Mathematical Surveys and Monographs. 92. Providence, RI: American Mathematical Society (AMS). xiv, 461 p. (2002).

This is a two-volume book that “represents a mixture of harmonic and complex analysis with operator theory. The interplay between these disciplines is one of the most significant features of the second part of twenthieth century mathematics. It gave rise to several jewels of analysis, such as the theory of singular integral operators, Toeplitz operators, mathematical scattering theory, Szökefalvi-Nagy-Foias model theory, the L. de Branges proof of the Bieberbach conjecture, as well as solving the principal interpolation problems in complex analysis and discovering the structural properties of function spaces (from Besov to Bergman)”. (From author’s introduction).

The book is intended for sufficiently prepared students as well as for experts. The necessary background consists of the Lebesgue measure and \(L_p\) spaces, elements of Fourier series and Fourier transforms, elementary holomorphic functions, Stone-Weierstrass theorem, and some basic concepts in functional analysis and topology.

The book consists of four parts, parts A and B form the first volume and parts C and D the second one. Each part consists of 3-9 chapters and each chapter ends with sections on “Exercises and further results” and “Notes and remarks”. A quite substantial portion of material is included in these exercises. Each part starts with a short foreword explaining its main ideas. Each volume ends with a Bibliography (the same in both volumes) counting 1053 positions, Author Index, Subject Index and Symbol Index.

We describe now shortly the content of the book. Part A – “An introduction to Hardy classes” – consists of 8 chapters. The topics considered are: the basic Wiener-Beurling-Helson theory of \(S\)-invariant subspaces in \(L^2(T,\mu)\), where \(S\) is the shift operator and \(\mu\) a finite nonnegative measure on the unit circle; an introduction to the weighted mean square polynomial approximation; description of the Hardy space theory in its classical setting: boundary values, Poisson and Cauchy representations, canonical Riesz-Smirnov factorization, etc.; an introduction to and exploitation of the class \(D\) of V. I. Smirnov (a subclass of the Nevanlinna class) applying it to obtain extended Phragmén-Lindelöf principles; the Helson-Szegő theorem on weighted Fourier series; an outline of the Hardy space theory in the half-plane and on the line; a quick introduction to Wiener’s linear time-invariant filtering; an application of Hardy space techniques to the Riemann \(\zeta\)-function.

Part B – “Hankel and Toeplitz operators” – consists of the following chapters: Hankel operators and their symbols; Compact Hankel operators; Applications to Nevanlinna-Pick interpolation; Essential spectrum. The first step: elements of Toeplitz operators; Essential spectrum. The second step: the Hilbert matrix and other Hankel operators; Hankel and Toeplitz operators associated with moment problem; Singular numbers of Hankel operators; Trace class Hankel operators; Inverse spectral problems, stochastic processes and one-sided invertibility.

Part C – “Model operators and free interpolation” – consists of three chapters. The basic function model; Elements of spectral theory in the language of characteristic function; Decompositions in invariant subspaces and free interpolation.

The last part D – “Analytic problems in linear systems control” – has the following chapters: Basic theory; First optimizations: multiplicity of the spectrum and DISC; Eigenvector decompositions, vector-valued exponentials, and squared optimization; A glance at bases of exponentials and reproducing kernels; A brief introduction to \(H^\infty\) control.

The book is intended for sufficiently prepared students as well as for experts. The necessary background consists of the Lebesgue measure and \(L_p\) spaces, elements of Fourier series and Fourier transforms, elementary holomorphic functions, Stone-Weierstrass theorem, and some basic concepts in functional analysis and topology.

The book consists of four parts, parts A and B form the first volume and parts C and D the second one. Each part consists of 3-9 chapters and each chapter ends with sections on “Exercises and further results” and “Notes and remarks”. A quite substantial portion of material is included in these exercises. Each part starts with a short foreword explaining its main ideas. Each volume ends with a Bibliography (the same in both volumes) counting 1053 positions, Author Index, Subject Index and Symbol Index.

We describe now shortly the content of the book. Part A – “An introduction to Hardy classes” – consists of 8 chapters. The topics considered are: the basic Wiener-Beurling-Helson theory of \(S\)-invariant subspaces in \(L^2(T,\mu)\), where \(S\) is the shift operator and \(\mu\) a finite nonnegative measure on the unit circle; an introduction to the weighted mean square polynomial approximation; description of the Hardy space theory in its classical setting: boundary values, Poisson and Cauchy representations, canonical Riesz-Smirnov factorization, etc.; an introduction to and exploitation of the class \(D\) of V. I. Smirnov (a subclass of the Nevanlinna class) applying it to obtain extended Phragmén-Lindelöf principles; the Helson-Szegő theorem on weighted Fourier series; an outline of the Hardy space theory in the half-plane and on the line; a quick introduction to Wiener’s linear time-invariant filtering; an application of Hardy space techniques to the Riemann \(\zeta\)-function.

Part B – “Hankel and Toeplitz operators” – consists of the following chapters: Hankel operators and their symbols; Compact Hankel operators; Applications to Nevanlinna-Pick interpolation; Essential spectrum. The first step: elements of Toeplitz operators; Essential spectrum. The second step: the Hilbert matrix and other Hankel operators; Hankel and Toeplitz operators associated with moment problem; Singular numbers of Hankel operators; Trace class Hankel operators; Inverse spectral problems, stochastic processes and one-sided invertibility.

Part C – “Model operators and free interpolation” – consists of three chapters. The basic function model; Elements of spectral theory in the language of characteristic function; Decompositions in invariant subspaces and free interpolation.

The last part D – “Analytic problems in linear systems control” – has the following chapters: Basic theory; First optimizations: multiplicity of the spectrum and DISC; Eigenvector decompositions, vector-valued exponentials, and squared optimization; A glance at bases of exponentials and reproducing kernels; A brief introduction to \(H^\infty\) control.

Reviewer: Wiesław Tadeusz Zelazko (Warszawa)

### MSC:

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

30-02 | Research exposition (monographs, survey articles) pertaining to functions of a complex variable |

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

47A40 | Scattering theory of linear operators |

47A57 | Linear operator methods in interpolation, moment and extension problems |

47A45 | Canonical models for contractions and nonselfadjoint linear operators |

47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |

43A15 | \(L^p\)-spaces and other function spaces on groups, semigroups, etc. |

30D55 | \(H^p\)-classes (MSC2000) |

93B05 | Controllability |

94C05 | Analytic circuit theory |

43A25 | Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups |