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Published online by Cambridge University Press: 25 June 2025
This survey is an introduction to the theory of Hankel operators, a beautiful area of mathematical analysis that is also very important in applications. We start with classical results: Kronecker's theorem, Nehari's theorem, Hartman's theorem, Adamyan-Arov-Krein theorems. Then we describe the Hankel operators in the Schatten-von Neumann class Sp and consider numerous applications: Sarason's commutant lifting theorem, rational approximation, stationary processes, best approximation by analytic functions. We also present recent results on spectral properties of Hankel operators with lacunary symbols. Finally, we discuss briefly the most recent results involving Hankel operators: Pisier's solution of the problem of similarity to a contraction, self-adjoint operators unitarily equivalent to Hankel operators, and approximation by analytic matrix-valued functions.
1. Introduction
I would like to invite the reader on an excursion into the theory of Hankel operators, a beautiful and rapidly developing domain of analysis that is important in numerous applications.
It was Hankel [1861] who began the study of finite matrices whose entries depend only on the sum of the coordinates, and therefore such objects are called Hankel matrices. One of the first theorems about infinite Hankel matrices was obtained by Kronecker [1881]; it characterizes Hankel matrices of finite rank. Hankel matrices played an important role in many classical problems of analysis, and in particular in the so-called moment problems; for example, Hamburger's moment problem is solvable if and only if the corresponding infinite Hankel matrix is positive semi-definite [Hamburger 1920; 1921].
Since the work of Nehari [1957] and Hartman [1958] it has become clear that Hankel operators are an important tool in function theory on the unit circle.
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