##
**Treatise on the shift operator. Spectral function theory. With an appendix by S. V. Khrushchev and V. V. Peller. Transl. from the Russian by Jaak Peetre.**
*(English)*
Zbl 0587.47036

Grundlehren der Mathematischen Wissenschaften, 273. Berlin etc.: Springer-Verlag. XI, 491 p. DM 184.00 (1986).

This edition has been considerably enlarged, as compared with the Russian original [Lektsii ob operatore sdviga (1980; Zbl 0508.47001)]. In particular, four entirely new appendices (Appendix 2-5) have been written.

A specific feature of the organization of the Russian edition (and hence of the present one) was its two-level structure (in the review cited even three levels were mentioned, but this does not seem to me quite correct). Each chapter (”Lecture”) was divided into two parts by the title ”Supplements and Bibliographical Notes”, the first parts constituting together an introduction to the subject and the rest ones being intended for a more advanced study. The enlargements made for the present edition contribute to both levels that makes the book more accessible for the beginners and more valuable for the advanced readers.

The main contribution to the introductory level is Appendix 2 ”Summary on \(H^ p\) spaces” that contains proofs of all assertions about Hardy classes used in the text.

Other additions and changes (we do not mention local improvements) contribute to the advanced level. In the main body of the book two alterations should be mentioned. Section 4 of Lecture II has been revised in the spirit of the paper [V. I. Vasyunin and N. K. Nikol’skij, Izv. Akad. Nauk SSSR, Ser. Mat., 47, 942-960 (1983)]. In Section 4 of Lecture 8 a new approach to the problem of free interpolation in \(H^{\infty}\) has been discussed (the interpolation operator of Jones-Vinogradov). We turn to reviewing Appendices 3 to 5.

Appendix 3 ”The Corona Problem, the Problem of Sz.-Nagy, Ideals in the Algebra \(H^{\infty}\)” is devoted to ”the modern proof of the Carleson Corona Theorem and its operator theoretic generalizations (Tom Wolff’s proof as modified by Tolokonnikov et al., with the best estimates for the solving functions known at present (1983))”. Related information on finitely generated ideals in \(H^{\infty}\) and a discussion of corona theorems in other Banach algebras are also included.

Appendix 4 ”Essays on the Spectral Theory of Hankel and Toeplitz Operators” is ”a rather extensive survey” of this theory ”connected with the general orientation of the book - the spectral function theory”. Main topics included: the Adamyan-Arov-Krein Theorem (the distances from a Hankel operator to the set of all rank n operators and to the set of all rank n Hankel operators are equal) and some material around it; spectra, essential spectra and explicit formulae for the index of Toeplitz operators for various classes of symbols; algebras generated by Toeplitz operators (symbolic calculus, the structure of the commutator ideal including questions of compactness of commutators and semicommutators, relations between the spectra of an operator and its symbol, etc.).

Appendix 5 ”Hankel Operator of Schatten-von Neumann Class and Their Applications to Stationary Processes and Best Approximations” has been written by V. V. Peller and S. V. Khrushchëv especially for this edition. It is based on the paper of the same authors [Usp. Mat. Nauk 37, No.1(233), 53-124 (1982; Zbl 0497.60033)], but in addition contains a proof of Peller’s theorem on characterization of Hankel operators of class \({\mathfrak S}_ p\) in terms of their symbols.

The author has indicated to me some misprints important for understanding (in statements of theorems, etc.) that are listed below:

p. 151. In the 11-th line from the bottom: ”\(\leq ''\) should be inserted between ”a” and ”\(| {\mathcal H}| ''.\)

p. 291. The 16-th line from the top should end by ”\(\leq \sqrt{2}\delta^{-2}\omega (\alpha,\beta)\Delta a''\) not \(''=\sqrt{2}\omega (\alpha,\beta)\Delta a''.\)

p. 353. In the box for \(\sigma_ x(Sym A)\), \(X=H^{\infty}+C\) of the table the symbol \(\lim_{| \zeta | \to 1}\) should be replaced by \(\underline{\lim}_{| \zeta | \to 1}.\)

p. 361. In the 13-th line from the bottom read ”deg \(P_-{\bar \phi}=0\) or deg \(P_-\psi =''\) instead of ”deg \(P_-\phi =0\) or deg \(P_-\psi =''.\)

p. 369. The 19-th line from the bottom should end by \(H_{{\bar \Theta}_ if}| K_{\Theta_ i}\), not \(H_{\Theta_ if_ i}.\)

p. 374. In the 4-th line from the top the middle term of the double inequality should be deleted.

A specific feature of the organization of the Russian edition (and hence of the present one) was its two-level structure (in the review cited even three levels were mentioned, but this does not seem to me quite correct). Each chapter (”Lecture”) was divided into two parts by the title ”Supplements and Bibliographical Notes”, the first parts constituting together an introduction to the subject and the rest ones being intended for a more advanced study. The enlargements made for the present edition contribute to both levels that makes the book more accessible for the beginners and more valuable for the advanced readers.

The main contribution to the introductory level is Appendix 2 ”Summary on \(H^ p\) spaces” that contains proofs of all assertions about Hardy classes used in the text.

Other additions and changes (we do not mention local improvements) contribute to the advanced level. In the main body of the book two alterations should be mentioned. Section 4 of Lecture II has been revised in the spirit of the paper [V. I. Vasyunin and N. K. Nikol’skij, Izv. Akad. Nauk SSSR, Ser. Mat., 47, 942-960 (1983)]. In Section 4 of Lecture 8 a new approach to the problem of free interpolation in \(H^{\infty}\) has been discussed (the interpolation operator of Jones-Vinogradov). We turn to reviewing Appendices 3 to 5.

Appendix 3 ”The Corona Problem, the Problem of Sz.-Nagy, Ideals in the Algebra \(H^{\infty}\)” is devoted to ”the modern proof of the Carleson Corona Theorem and its operator theoretic generalizations (Tom Wolff’s proof as modified by Tolokonnikov et al., with the best estimates for the solving functions known at present (1983))”. Related information on finitely generated ideals in \(H^{\infty}\) and a discussion of corona theorems in other Banach algebras are also included.

Appendix 4 ”Essays on the Spectral Theory of Hankel and Toeplitz Operators” is ”a rather extensive survey” of this theory ”connected with the general orientation of the book - the spectral function theory”. Main topics included: the Adamyan-Arov-Krein Theorem (the distances from a Hankel operator to the set of all rank n operators and to the set of all rank n Hankel operators are equal) and some material around it; spectra, essential spectra and explicit formulae for the index of Toeplitz operators for various classes of symbols; algebras generated by Toeplitz operators (symbolic calculus, the structure of the commutator ideal including questions of compactness of commutators and semicommutators, relations between the spectra of an operator and its symbol, etc.).

Appendix 5 ”Hankel Operator of Schatten-von Neumann Class and Their Applications to Stationary Processes and Best Approximations” has been written by V. V. Peller and S. V. Khrushchëv especially for this edition. It is based on the paper of the same authors [Usp. Mat. Nauk 37, No.1(233), 53-124 (1982; Zbl 0497.60033)], but in addition contains a proof of Peller’s theorem on characterization of Hankel operators of class \({\mathfrak S}_ p\) in terms of their symbols.

The author has indicated to me some misprints important for understanding (in statements of theorems, etc.) that are listed below:

p. 151. In the 11-th line from the bottom: ”\(\leq ''\) should be inserted between ”a” and ”\(| {\mathcal H}| ''.\)

p. 291. The 16-th line from the top should end by ”\(\leq \sqrt{2}\delta^{-2}\omega (\alpha,\beta)\Delta a''\) not \(''=\sqrt{2}\omega (\alpha,\beta)\Delta a''.\)

p. 353. In the box for \(\sigma_ x(Sym A)\), \(X=H^{\infty}+C\) of the table the symbol \(\lim_{| \zeta | \to 1}\) should be replaced by \(\underline{\lim}_{| \zeta | \to 1}.\)

p. 361. In the 13-th line from the bottom read ”deg \(P_-{\bar \phi}=0\) or deg \(P_-\psi =''\) instead of ”deg \(P_-\phi =0\) or deg \(P_-\psi =''.\)

p. 369. The 19-th line from the bottom should end by \(H_{{\bar \Theta}_ if}| K_{\Theta_ i}\), not \(H_{\Theta_ if_ i}.\)

p. 374. In the 4-th line from the top the middle term of the double inequality should be deleted.

Reviewer: S.Kisljakov

### MSC:

47B37 | Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) |

47-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operator theory |

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

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 |

46J15 | Banach algebras of differentiable or analytic functions, \(H^p\)-spaces |