Abstract
A uniform algebra A on a compact space X is tight if for each g ϵ C( X), the Hankel-type operator f → gf + A from A to C 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 C n for which a certain ∂ -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 ⊥.
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