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