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Shift invariant subspaces in the Bloch space

Adem Limani, Artur Nicolau

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.

Shift invariant subspaces in the Bloch space

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.
Paper Structure (18 sections, 32 theorems, 159 equations)

This paper contains 18 sections, 32 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Beurling-type theorems and model spaces
  3. Regular spaces
  4. $M_z$-invariant subspaces in duals of Regular Spaces
  5. Proof of \ref{['THM:Modelspaces']} in the context of Regular Spaces
  6. An abstract decomposition for singular measures
  7. The permanence property and boundary zero sets
  8. Inner factors in weighted BMOA
  9. The permanence property and $W^1$
  10. The permanence property and invisibility
  11. Singular inner functions in $W^1$
  12. No estimate from below implies cyclicity in $\mathcal{B}$
  13. The permanence property discriminates no compact set
  14. Cyclicity and invertibility
  15. A sufficient condition for cyclicity
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $f=FS_\mu B$ be in $H^\infty$, where $F$ is outer, $S_\mu$ is singular inner and $B$ is a Blaschke product. Then there exists a unique (up to sets of $\mu$-measure zero) decomposition $\mu = \mu_P + \mu_C$ of $\mu$, where $\mu_P, \mu_C$ are mutually singular positive measures giving rise to the

Theorems & Definitions (56)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 46 more