Do some nontrivial closed $z$-invariant subspaces have the division property?

Abstract:

Banach spaces $E$ of functions holomorphic on the open unit disk $\mathbb {D}$ are considered such that the unilateral shift $S$ and the backward shift $T$ are bounded on $E$. Under the assumption that the spectra of $S$ and $T$ are equal to the closed unit disk, the existence is discussed of closed $z$-invariant subspaces $N$ of $E$ having the “division property,” which means that the function $f_{\lambda }\colon z \mapsto \frac {f(z)}{z-\lambda }$ belongs to $N$ for every $\lambda \in \mathbb {D}$ and for every $f \in N$ with $f(\lambda )=0$. This question is related to the existence of nontrivial bi-invariant subspaces of Banach spaces of hyperfunctions on the unit circle $\mathbb {T}$.