An \(L^{p}\) analog to AAK theory for \(p\geq 2\)
This paper deals with Hankel operators from the \(H^ s \) space of the unit circle into \({\overline H}^2_0 \), where \(1/p+1/s=1/2 \). Interpolating between the two natural cases \(p=2 \) and \(p=\infty \), the authors develop an \(L^ p \) analogue of the Adamyan-Arov-Kreĭn theory. In particular, they show that the distance to the set of operators of rank at most \(m \) is equal to the distance to the set of Hankel operators of rank at most \(m \). They also explain the relationship between the \(m\)-th singular value and the best approximation by meromorphic functions with at most \(m \) poles in the disk. Various applications to rational approximation are also given. There are related papers by \textit{C. Le Merdy} [Bull. Lond. Math. Soc. 25, 275--281 (1993; Zbl 0793.47024)] and by \textit{V. A. Prokhorov} [J. Approximation Theory 116, 380--396 (2002; Zbl 1017.47024)]. For the extensive theory of Hankel operators and related topics, we refer the reader to the monographs by \textit{S.~C.~Power} [``Hankel operators on Hilbert space'' (Research Notes in Mathematics 64, Pitman) (1982; Zbl 0489.47011)] and \textit{J.~R.~Partington} [``An introduction to Hankel operators'' (London Mathematical Society Student Texts 13, Cambridge University Press) (1988; Zbl 0668.47022)], and especially to the recent one by \textit{V.V.~Peller} [``Hankel operators and their applications'' (Springer Monographs in Mathematics, Springer Verlag, New York) (2003; Zbl 1030.47002)]. The latter contains a detailed account of the theory, including a summary of some results of the paper under review (see Chapter~4, Concluding Remarks on pp. 145--146). The paper under review contains a wealth of material. The list of sections describes very well the content: 1. Introduction, 2. Analytic Approximation, 3. A Hankel Operator Approach, 4. Extremal Functions and Maximizing Vectors, 5. Meromorphic Approximation, 6. Interpolation Between \(p=2 \) and \(p=\infty \), 7. Some Ljusternik-Schnirelman (Lyusternik-Shnirel'man) Theory, 8. AAK Theory in \(L^ p \) for \(p\geq 2 \), 9. Critical Points, 10. Nonhermitian Orthogonality. The bibliography contains more than 60 items and is rather complete. Regarding those results in the paper that deal with the best approximation, it might be worth also adding as a reference the well-known monograph by \textit{H. Shapiro} [Topics in approximation theory, Lecture Notes in Mathematics, 187, Springer-Verlag (1971; Zbl 0213.08501)].
scientific article; zbMATH DE number 1797584
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