Tight uniform algebras and algebras of analytic functions

A uniform algebra A on a compact space X is tight if for each g?C(X), the Hankel-type operator f → gf + A from $\text{A}\text{to}\text{C}\text{A}$ is weakly compact. Two families of uniform algebras are shown to be tight: the algebras such as R(K) that arise in the theory of rational approximation on compact subsets of the complex plane, and algebras of analytic functions on domains in $\text{C}$ⁿ for which a certain $\text{?}$-problem is solvable. A couple of characterizations of tight algebras are given, and one of these is used to show that the property of being tight places severe restrictions on the Gleason parts of A and the measures in A^⊥.