The duals to the spaces of analytic vectorvalued functions and the duality of functions, generated by these spaces

A. V. Bukhvalov

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part IX, Tome 92 (1979), pp. 30-50

Résumé

Let $X$ a complex Banach space, $1$, $1/p+1/p'=1$; $A^p(X)$ is the space of all $X$-valued analytic in the open disk $L^p$-integrable functions. By means of the natural duality it is proved that $A^p(X)^*=A^{p'}(X^*)$. Let $\mathbf A^p$ and $\mathbf H^p$ be the functors in a category of Banach spaces, generated by $A^p(X)$ and the Hardy space $H^p(X)$ respectively. With some restrictions on the category the following it true: 1) $D\mathbf A^p=\mathbf A^{p'}$; 2) $H^p(X)^*=D\mathbf H^p(X^*)$; 3) $D\mathbf H^p\ne\mathbf H^{p'}$ in the category of all separable reflexive Banach spaces; 4) the functors $\mathbf A^p$ and $\mathbf H^p$ are reflexive.