Shift invariant subspaces of large index in the Bloch space
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929488288284672 |
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| author | Biehler, Nikiforos |
| author_facet | Biehler, Nikiforos |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_03562 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Shift invariant subspaces of large index in the Bloch space Biehler, Nikiforos Functional Analysis Complex Variables 30H30, 30B10, 47A15, 47B91 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. |
| title | Shift invariant subspaces of large index in the Bloch space |
| topic | Functional Analysis Complex Variables 30H30, 30B10, 47A15, 47B91 |
| url | https://arxiv.org/abs/2409.03562 |