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

Abstract

We consider Banach spaces E of functions holomorphic on the open unit disc D such that the unilateral shift S and the backward shift T are bounded on E. Assuming that the spectra of S and T are equal to the closed unit disc we discuss the existence of closed z-invariant of N of E having the "division property", which means that the function f $\lambda$ : z $\to$ f (z)/ z--$\lambda$ belongs to N for every $\lambda$ $\in$ D and for every f $\in$ N such that 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.