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 functionz−λ belongs to N for every λ ∈ D and for every f ∈ N such that f (λ) = 0. This question is related to the existence of nontrivial bi-invariant subspaces of Banach spaces of hyperfunctions on the unit circle T.