Bonded projections, duality, and multipliers in spaces of analytic functions

Abstract

Let q) and b be positive continuous functions on [0, 1) with T(r) -*0 as r -11 and f S 0(r) dr<xo. Denote by Ao(qp) and Ao(q)) the Banach spaces of functionsfanalytic in the open unit disc D with 1f(z)1q(z1)=o(1) and 1f(z)1p(1z1)=O(1), lzl -1, respectively. In both spaces 1IfIL,=supD Jf(z)Jp(Jz1). Let A'(1) denote the space of functions analytic in D with lffl1,, =JfJD lf(z)kb(IzI) dx dy < oc. The spaces Ao(q)), A'(b), and Ao(q)) are identified in the obvious way with closed subspaces of Co(D), L'(D), and L-(D), respectively. For a large class of weight functions q, which go to zero at least as fast as some power of (1 r) but no faster than some other power of (1 -r), we exhibit bounded projections from Co(D) onto Ao(q), from L'(D) onto A'(b), and from L'(D) onto Ao(q). Using these projections, we show that the dual of Ao(q) is topologically isomorphic to A'(/) for an appropriate, but not unique choice of 0. In addition, Ao(qp) is topologically isomorphic to the dual of A'(/). As an application of the above, the coefficient multipliers of Ao(q), A'(0), and A. (q)) are characterized. Finally, we give an example of a weight function pair q), 0 for which some of the above results fail. We study certain Banach spaces of analytic functions in the unit disc D. For example, let A,, denote the space of functions satisfying If(z)I=O((1-IzI)-1), let Ao denote the subspace with 0 replaced. by o, and let A1 denote the space of functions with ff If dx dy < oo. We may regard AO as a subspace of Co(D), the continuous functions on the disc that vanish on the boundary. We first show that there are bounded projections from Co(D) onto AO, from L1(D) onto A1, and from L'(D) onto A,, (Theorem 1). Using this, Lindenstrauss and Pelczynski have shown that A1 is topologically isomorphic to 11. Using this we show in Theorem 2 that A1 may be identified with the conjugate space of AO, and A,, with the conjugate space of A1. (For the special case indicated above this result is in Duren, Romberg, and Shields [2, Theorems 7 and 11], but our proof applies to more general weight functions, as indicated below.) Finally, we find all sequences {An} such that 2 Anazn E A,, whenever 2 anzn E A, and likewise for Ao and Al (Theorem 3). Presented to the Society, January 22, 1970 under the title Spaces of analytic functions with weighted supremum niorms; received by the editors July 22, 1970 and, in revised form, March 1, 1971. AMS 1970 subject classifications. Primary 46E15; Secondary 30A98.