The second duals of certain spaces of analytic functions

Abstract

Let <p be a continuous, decreasing, real-valued function on 0 ^ r g 1 with 95(1) = 0 and <p(r) > 0 for r < 1. Let E o be the Banach space of analytic functions / on the open unit disc D, such that f(z)<p(\z\) -> 0 as z\ -> 1, with norm \\f\\ = svv{\f(z)}<p(z):zeD}, where we write tp(z) = (p(\z\) for z e D. Let E be the Banach space of analytic functions f on D for which ftp is bounded in D, with the same norm as E o . It is easy to see that E is complete in this norm, and that E o is a closed subspace of E. The dual space of E o will be shown to be identifiable with a quotient space of L X {D). Hence the second dual can be identified with a subspace of L°°(D). Our main result is that E may be naturally identified with this second dual, and that the inclusion map of E o into E coincides with the natural embedding of E o in E$*. The corresponding result fails, as one can easily see, if the hypothesis of analyticity is replaced by mere continuity in the definitions of E o and E. The result bears some similarity to the situation for sequence spaces; the second dual of the space of null sequences is the space of bounded sequences. It should also be compared with results of de Leeuw [2] and of Duren, Romberg, and Shields [3, § 4] for spaces of Lipschitzian functions. Let q)E 0 = {cpf : / e E o }; q>E is defined similarly. Let Here, dA denotes two-dimensional Lebesgue measure on D. If geL 1 {D), let [g] denote the coset g+N 1 that contains g. Thus, [g] is an element of the

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