An Excursion into the Theory of Hankel Operators

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.

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