Hankel-Type Operators, Bourgain Algebras, and Uniform Algebras

Let H∞(D) denote the algebra of bounded analytic functions on the open unit disc in the complex plane. For a function g ∈ L∞(D), the Hankel-type operator Sg is defined by Sg(f) = gf +H∞(D). We give here an overview of the study of the symbol of the Hankel-type operator, with emphasis on those symbols for which the operator is compact, weakly compact, or completely continuous. We conclude with a look at this operator on more general domains and several open questions. We look at a uniform algebra A on a compact Hausdorf space X. We let M(A) denote the maximal ideal space of A. We will consider the Hankel-type operator Sg : A → C(X)/A with symbol g ∈ C(X) defined by Sg(f) = fg + A for all f ∈ A. Even though the space L does not look like an algebra of continuous functions, it is possible to identify it with the space of continuous functions on its maximal ideal space X as follows: for f in L define the Gelfand transform of f by f̂(x) = x(f) for all x ∈ X. Since the topology on X is given by saying that a net xα converges to x in X if and only if xα(f) converges to x(f) for all f ∈ L, we see that the Gelfand transform defines a continuous function on X. We will be most interested in the case in which A = H(U), the algebra of bounded analytic functions on a bounded domain U in the complex plane, and C(X) = L(U) with respect to area measure. When the domain does not matter or when we think no confusion should arise, we will write simply H or L. The purpose of this article is to indicate why we look at such operators, what one can do with these Hankel-type operators, and some of what remains to be done in this area.