An Introduction to Hankel Operators

Abstract

In this introductory chapter we define the Hankel operators and study their basic properties. We introduce in §1 the class of Hankel operators as operators with matrices of the form ${\left\{ {{\alpha _{i}} + k} \right\}_{{j,k}}} \geqslant 0$ and consider different realizations of such operators. One of the most important realization is the Hankel operators H _φ , from the Hardy class $H_{-}^{2}\mathop{ = }\limits^{{def}} {L^{2}} \odot {H^{2}}$. We prove the fundamental Nehari theorem, which describes the bounded Hankel operators, and we discuss the problem of finding symbols of minimal norm. We introduce the important Hilbert matrix, prove its boundedness, and estimate its norm. Then we study Hankel operators with unimodular symbols. We conclude §1 with the study of commutators of multiplication operators with the Riesz projection on L² and reduce the study of such commutators to the study of Hankel operators.