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arXiv · 2409.03562

Shift invariant subspaces of large index in the Bloch space

Nikiforos Biehler
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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.

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Nikiforos Biehler. 2024-09-05. Shift invariant subspaces of large index in the Bloch space. https://arxiv.org/abs/2409.03562

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