Sub-Hardy Hilbert spaces on the circle and torus

Sahni and Singh settled a problem posed by Yousefi & Hesameddini by generalizing their main result with a simple proof in the setting of sub-Hilbert spaces in $H^2(T)$. In this paper, we extend the main result of Sahni and Singh to the setting of the Banach spaces $H^p$, $1\le p\le\infty$ on the circle and the torus.