Having established the necessary background, we cover a variety of topics in this chapter. Starting from the definition of a Hankel matrix we give three equivalent approaches to the task of defining a Hankel operator on H2 – that is, an operator whose matrix is a Hankel matrix with respect to the usual basis, {1, z, z2, …}. All three approaches have been used in the literature and we choose what is in some ways the simplest, explaining how one can easily pass from this to the others.

The first big theorem is Nehari's theorem, which associates with a bounded Hankel operator a function in L∞(T) (a symbol) whose norm is the same as the operator norm. We give some examples, including Hilbert's Hankel matrix.

Next we come to two famous problems of complex analysis, the Carathéodory-Fejér and Nevanlinna-Pick problems, which we state in their simplest forms (many more difficult versions have been analysed). Each can be reduced to the Nehari extension problem – that of finding a symbol of minimum norm – and we give Sarason's elegant solution to this.

Turning to the more general theory of Hankel operators, we give Kronecker's theorem, characterising finite-rank Hankel operators in terms of rational symbols. We then prove Hartman's theorem characterising compact Hankel operators – a much deeper result. En route we introduce the disc algebra, a subspace of H∞.

Much of the material of this chapter is covered by the cited works of Francis, Garnett, Power and Sarason.