Shift invariant subspaces of large index in the Bloch space

TL;DR

This paper investigates the structure of shift-invariant subspaces in the Bloch and little Bloch spaces, constructing examples with arbitrarily large index and analyzing stability properties under weak-star topology.

Contribution

It introduces methods to construct shift-invariant subspaces with large index and establishes stability results for the index in the weak-star topology.

Findings

Constructed shift-invariant subspaces with index up to the cardinality of [0,1]

Proved existence of weak-star closed invariant subspaces with arbitrary large index

Established stability of the index under weak-star closure in Banach spaces

Abstract

We consider the shift operator $M_z$, defined on the Bloch space and the little Bloch space and we study the corresponding lattice of invariant subspaces. The index of a closed invariant subspace $E$ is defined as $\text{ind}(E) = \dim(E/M_z E)$. We construct closed, shift invariant subspaces in the Bloch space that can have index as large as the cardinality of the unit interval $[0,1]$. Next we focus on the little Bloch space, providing a construction of closed, shift invariant subspaces that have arbitrary large index. Finally we establish several results on the index for the weak-star topology of a Banach space and prove a stability theorem for the index when passing from (norm closed) invariant subspaces of a Banach space to their weak-star closure in its second dual. This is then applied to prove the existence of weak-star closed invariant subspaces of arbitrary index in the Bloch space.