Let $H^2(\Delta^2)$ denote the Hardy space on the polydics and $M_f:H^2(\Delta^2)\rightarrow H^2(\Delta^2)$ be a multiplication operator by $f\in H^\infty(\Delta^2)$. Is the operator is bijective?
Actually, I want to show that if the multiplication operator above has closed range, then $f^{-1}\in L^\infty(T^2)$. Is it true?