Bounded compact and dual compact approximation properties of Hardy spaces: New results and open problems

The aim of the paper is to highlight some open problems concerning approximation properties of Hardy spaces. We also present some results on the bounded compact and the dual compact approximation properties (shortly, BCAP and DCAP) of such spaces, to provide background for the open problems. Namely, we consider abstract Hardy spaces $H\left[X\left(w\right)\right]$ built upon translation-invariant Banach function spaces $X$ with weights $w$ such that $w\in X$ and $w^{-1}\in X^{\prime }$, where $X^{\prime }$ is the associate space of $X$. We prove that if $X$ is separable, then $H\left[X\left(w\right)\right]$ has the BCAP with the approximation constant $M\left(H\left[X\left(w\right)\right]\right)\le 2$. Moreover, if $X$ is reflexive, then $H\left[X\left(w\right)\right]$ has the BCAP and the DCAP with the approximation constants $M\left(H\left[X\left(w\right)\right]\right)\le 2$ and $M^{\ast }\left(H\left[X\left(w\right)\right]\right)\le 2$, respectively. In the case of classical weighted Hardy space $H^p\left(w\right)=H\left[L^p\left(w\right)\right]$ with $1<p<\infty$, one has a sharper result: $M\left(H^p\left(w\right)\right)\le 2^{|1-2/p|}$ and $M^{\ast }\left(H^p\left(w\right)\right)\le 2^{|1-2/p|}$.