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

Abstract

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