Shift invariant subspaces of large index in the Bloch space

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. We construct closed, shift invariant subspaces in the Bloch space and the little Bloch space that can have arbitrarily large, but countable, index. On the non-separable Bloch space we construct a closed shift invariant subspace with cardinality equal to the unit interval. 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.