Shift invariant subspaces in the Bloch space

TL;DR

This paper investigates the structure of shift-invariant subspaces in the Bloch space, revealing new decompositions involving singular inner functions and contrasting with Bergman space results, thus answering longstanding open questions.

Contribution

It introduces a novel decomposition of bounded analytic functions in the Bloch space involving cyclic and shift-invariant factors, highlighting differences from Bergman space theory.

Findings

Decomposition of functions into cyclic and shift-invariant factors using singular inner functions.

Shift-invariant subspaces in the Bloch space differ significantly from those in Bergman spaces.

Existence of invertible functions in the Bloch space that are not cyclic.

Abstract

We consider weak-star closed invariant subspaces of the shift operator in the classical Bloch space. We prove that any bounded analytic function decomposes into two factors, one which is cyclic and another one generating a proper shift invariant subspace, satisfying a permanence property, which in a certain way is opposite to cyclicity. Singular inner functions play the crucial role in this decomposition. We show in several different ways that the description of shift invariant subspaces generated by inner functions in the Bloch spaces deviates substantially from the corresponding description in the Bergman spaces, provided by the celebrated Korenblum and Roberts Theorem. Furthermore, the relationship between invertibility and cyclicity is also investigated and we provide an invertible function in the Bloch space which is not cyclic therein. Our results answer several open questions stated in the early nineties.