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Hankel operators and their applications. (English) Zbl 1030.47002
Springer Monographs in Mathematics. New York, NY: Springer. xv, 784 p. EUR 89.95/net; sFr. 149.50; £63.00; \$ 99.95 (2003).
Let $$\mathbf T$$ be the unit circle, $$H^2$$ the Hardy subspace of all functions in $$L^2(\mathbf T)$$ whose negative Fourier coefficients vanish, i.e., $$\int_0^{2\pi} f(e^{it}) e^{nit} dt=0$$ for all $$n>0$$, and $$H^2_-$$ the orthogonal complement of $$H^2$$ in $$L^2(\mathbf T)$$. For $$\varphi\in L^\infty(\mathbf T)$$, the Hankel operator with symbol $$\varphi$$ is the operator from $$H^2$$ into $$H^2_-$$ defined by $$H_\varphi f=P_-(\varphi f)$$, where $$P_-:L^2(\mathbf T)\to H^2_-$$ is the orthogonal projection. It turns out that Hankel operators are precisely those whose matrix with respect to the standard orthonormal bases $$\{e^{kit}\}_k$$ of $$H^2$$ and $$H^2_-$$ have constant entries along each diagonal perpendicular to the main diagonal, i.e., the $$(i,j)$$-entry depends only on $$i+j$$. These operators have proved a fascinating object of study for the last 40 years or so, and have found numerous applications in operator theory, optimal control, Gaussian processes, and elsewhere.
The present monograph, comprising of almost 800 pages, is an overwhelmingly concise and lucid exposition of the whole theory by one of the leading contributors to the field. The only topics which are not included are the various generalizations of Hankel operators to Bergman spaces or to domains in $$\mathbb{C}^n$$; however, this is understandable and even wise, as the latter theories have a completely different flavour and, above all, use very different tools. A good reference for these is K. Zhu’s monograph [“Operator theory in function spaces” (Pure Appl. Math. 139, Marcel Dekker, New York) (1990; Zbl 0706.47019)].
The book begins with very basic material (Chapter 1), mostly going back to the late 1950s and 1960s – the Nehari theorem, connections between Hankel operators and BMO and VMO, the Sz.-Nagy-Foias functional model, the Nevanlinna-Pick problem, Carleson measures, etc. The next two chapters provide similar basic information on vectorial Hankel operators and on the companion Toeplitz (and vectorial Toeplitz) operators, respectively. Chapter 4 focuses on the Adamian-Arov-Krein theorem and singular values of Hankel operators. Chapter 5 describes the various parametrizations of the solutions of the Nehari problem, as well as of the related interpolation problems (Nevanlinna-Pick etc.). The next chapter gives an account of the results on the membership of $$H_\varphi$$ in the Schatten classes, and on rational approximation in BMO, obtained by the author in the early 1980s [Dokl. Akad. Nauk SSSR 252, 43-48 (1980; Zbl 0484.47011); Mat. Sb. 113(155), 538-581 (1980; Zbl 0458.47022)], and later complemented by Semmes, Pekarski, and many others. The topic of approximation by rational, analytic or meromorphic functions is then taken further in the ensuing Chapter 7, treating also in detail various particular function spaces (Besov spaces, Hölder-Zygmund classes, etc.). Chapters 8 and 9 feature applications to stationary Gaussian processes (initiated by the author and Khrushchev in mid-1980s), the Helson-Szegö theorem, and related issues. Chapter 10 deals with the spectra of Hankel operators; this subject is then taken up in Chapter 12, which describes the solution of the inverse spectral problem for selfadjoint Hankel operators (i.e., to characterize bounded selfadjoint operators that are unitarily equivalent to a Hankel operator), given by the author, A. V. Megretskii and S. R. Treil [Acta Math. 174, 241-309 (1995; Zbl 0865.47015)]. This chapter also makes use of some methods from control theory; the applications of Hankel operators in the latter are discussed in the preceding Chapter 11.
Chapters 13 and 14 are again devoted to the topic of approximation. The former describes the solution to the matrix-valued recovery problem, i.e., describes under what conditions one can conclude from $$P_- f\in X$$ that $$f\in X$$, for a given function space $$X$$; this is tantamount to a certain hereditarity property of the Wiener-Hopf factorization of functions in $$X$$. The latter chapter gives an account of the theory of superoptimal approximations of matrix valued functions, initiated by N. J. Young and the author [J. Funct. Anal. 120, 300-343 (1994; Zbl 0808.47011)], with later advances by Treil and Alekseev. Finally, the last Chapter 15 concludes with an application of Hankel operators to the solution, due to G. Pisier [J. Am. Math. Soc. 10, 351-369 (1997; Zbl 0869.47014)], of the celebrated problem whether every polynomially bounded operator on a Hilbert space must be similar to a contraction.
There are two appendices, one on the general theory of operators on a Hilbert space (including, however, unitary dilations, the Sz.-Nagy-Foias functional model, and direct integrals), and one on the various function spaces that occur in the book (Hardy spaces, corona theorem, interpolating sequences, BMO and VMO, Besov spaces). These are followed by 37 pages of bibliography, as well as an author and a (rather rudimentary) subject index.
There are several older good books on Hankel operators or related fields, e.g., by R. Douglas [“Banach algebra techniques in operator theory” (Academic Press) (1972; Zbl 0247.47001)], S. C. Power [“Hankel operators on Hilbert space” (Pitman) (1982; Zbl 0489.47011)], N. K. Nikolskii [“Treatise on the shift operator” (Springer) (1986; Zbl 0587.47036)], J. R. Partington [“An introduction to Hankel operators” (Cambridge Univ. Press) (1988; Zbl 0668.47022)], or A. Böttcher and B. Silbermann [“Analysis of Toeplitz operators” (Akademie-Verlag) (1989; Zbl 0732.47029)], as well as the recent two-volume set by N. K. Nikolskii [“Operators, functions, and systems: an easy reading” (AMS) (2002; Zbl 1007.47001, Zbl 1007.47002)]; but none of them can probably match this excellent monograph in terms of depth of the treatment and of amount of the included material relevant to Hankel operators and their applications. The book has a very good chance of becoming one of the standard references on Hankel operators for the next decades.

##### MSC:
 47-02 Research exposition (monographs, survey articles) pertaining to operator theory 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
##### Keywords:
Hankel operators; Hardy space