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

Abstract:

Let $\varphi$ and $\psi$ be positive continuous functions on $[0,1)$ with $\varphi (r) \to 0$ as $r \to 1$ and $\smallint _0^1\psi (r)\;dr < \infty$. Denote by ${A_0}(\varphi )$ and ${A_\infty }(\varphi )$ the Banach spaces of functions f analytic in the open unit disc D with $|f(z)|\varphi (|z|) = o(1)$ and $|f(z)|\varphi (|z|) = O(1),|z| \to 1$, respectively. In both spaces $\left \|f\right \|_\varphi = {\sup _D}|f(z)|\varphi (|z|)$. Let ${A^1}(\psi )$ denote the space of functions analytic in D with $\left \|f\right \|_\psi = \smallint {\smallint _D}|f(z)|\psi (|z|)\;dx\;dy < \infty$. The spaces ${A_0}(\varphi ),{A^1}(\psi )$, and ${A_\infty }(\varphi )$ are identified in the obvious way with closed subspaces of ${C_0}(D),{L^1}(D)$, and ${L^\infty }(D)$, respectively. For a large class of weight functions $\varphi ,\psi$ 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 ${C_0}(D)$ onto ${A_0}(\varphi )$, from ${L^1}(D)$ onto ${A^1}(\psi )$, and from ${L^\infty }(D)$ onto ${A_\infty }(\varphi )$. Using these projections, we show that the dual of ${A_0}(\varphi )$ is topologically isomorphic to ${A^1}(\psi )$ for an appropriate, but not unique choice of $\psi$. In addition, ${A_\infty }(\varphi )$ is topologically isomorphic to the dual of ${A^1}(\psi )$. As an application of the above, the coefficient multipliers of ${A_0}(\varphi ),{A^1}(\psi )$, and ${A_\infty }(\varphi )$ are characterized. Finally, we give an example of a weight function pair $\varphi ,\psi$ for which some of the above results fail.