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

Abstract

Let <p and /i be positive continuous functions on [0, 1) with <p(r) ->-0 as r -> 1 and JJ ifi(r) dr <co. Denote by A0(<p) and A"(f) the Banach spaces of functions/analytic in the open unit disc D with |/(z)|y(|z|) = i?(l) and \f(z)\<p(\z\) = 0(\), \z\ -> 1, respectively. In both spaces \\f\\9 = supD |/(z)|ip(|z|). Let A1^) denote the space of functions analytic in D with ||/IL=JTd \f(,z)\<K\z\)dx dy<<x>. The spaces ^o(y), A1^), and A,(<p) are identified in the obvious way with closed subspaces of C0(D), H-D), and L"(D), respectively. For a large class of weight functions <p, </i 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 C0(D) onto A0(<p), from L1(D) onto A1^), and from L"(D) onto A"(<p). Using these projections, we show that the dual of A0(<p) is topologically isomorphic to A1^) for an appropriate, but not unique choice of <i. In addition, A"(y) is topologically isomorphic to the dual of /4'(0). As an application of the above, the coefficient multipliers of A0(<p), A\>ji), and Aa>((p) are characterized. Finally, we give an example of a weight function pair <P, i/i for which some of the above results fail.