Norm-compact sets of representing measures

Abstract

Let K be a compact subset of the complex plane C and let p be a point of KO, the interior of K. Let R(K) be the uniform closure on K of the rational functions with poles off K and let Mp be the set of all positive measures X on AK, the topological boundary of K, with the property that f8K4dX =4)(p) for all 4) in R(K). (Such a measure is called a representing measure for evaluation at p.) In an effort to cast some light on the problem of putting analytic structure in the maximal ideal space of a function algebra, Bishop has conjectured in [1, problem 8, p. 347] that Mp is compact in the norm topology as a subset of the space measures on AK. Theorem 1 of this paper shows that this is indeed the case for many compact sets; however, Theorem 2 gives some necessary conditions that Mp be norm-compact and consequently provides a number of counterexamples to Bishop's conjecture. If the compact set has only a finite number of components in its complement, then Mp is norm-compact. This follows immediately from the fact that the linear span of the real measures that annihilate R(K) is finite-dimensional. (This is a consequence of a classical theorem of Walsh [6, p. 518].) However, when K has infinitely many complementary components this argument is no longer valid, for both Mp and the space of real annihilating measures maycontain infinitely many linearly independent elements. For example, let K be the set obtained by deleting from the closed unit disc a sequence Ci } of open subdiscs with disjoint closures whose centers lie on the positive real axis and increase to 1 and whose radii decrease to 0. Let ,u be harmonic measure on AK for p and let fi be the element of LZ(OK, ,u) such that for each g in L1(OK, ,u), faKgf,dg is the period about C, of the harmonic conjugate of the harmonic extension of g to KO. It is easily seen thatf1,f2, * * * are linearly independent and consequently that the representing measures X= (1 (filc)),g are linearly independent, where c,= jlfiljl. Nevertheless, Mp is normcompact for this and other compact sets. In order to prove this we must make an assumption about two subsets of OK; these two subsets are defined below. Let K be compact and let Co, Ci, * * * be the components of C-K. For each integer n, n=O, 1, 2, . . . , let F,, consist of those points of