Minimality of the Riesz projection among projections onto abstract Hardy spaces and related topics

Let $X$ be a translation-invariant Banach function space on the unit circle<br/>$\mathbb{T}$ with the associate space $X'$, let $w$ be a weight such that <br/>$w\in X$ and $1/w\in X'$, let $X(w)$ consist of measurable functions<br/>$f:\mathbb{T}\to\mathbb{C}$ such that $fw\in X$, and let $H[X]$ and $H[X(w)]$<br/>denote the abstract Hardy spaces built upon $X$ and $X(w)$, respectively.<br/>Extending Rudin's arguments (1962), we show that if $\mathcal{P}$ is a bounded<br/>projection from $X(w)$ onto <br/>$H[X(w)]$, then the Riesz projection $P$ is bounded from $X$ onto $H[X]$ and <br/>$\|aI+bP\|_{\mathcal{B}(X)}\le \|aI+b\mathcal{P}\|_{\mathcal{B}(X(w))}$<br/>for all $a,b\in\mathbb{C}$. Further, for $m\in\mathbb{N}$, let <br/>$T(\mathbf{e}_{-m})$ be the Toeplitz operator with symbol <br/>$\mathbf{e}_{-m}(t)=t^{-m}$. We prove that <br/>$\|T(\mathbf{e}_{-m})\|_{\mathcal{B}(H[X])}<br/>\le\|T(\mathbf{e}_{-m})\|_{\mathcal{B}(H[X(w)])}$ for all $m\in\mathbb{N}$.