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Revisiting cyclic elements in growth spaces

Linus Bergqvist, Adem Limani, Bartosz Malman

TL;DR

The paper characterizes cyclic Nevanlinna functions for the shift on radial growth spaces $A^p(W)$ with finite Nevanlinna characteristic, treating both the slow (Dini-regular) and fast (logarithmic divergence) regimes for good weights. It reduces cyclicity to the singular inner factor, then develops a Roberts-type decomposition adapted to $W$ and provides a constructive, corona-free approach to realize cyclicity via explicit bounded holomorphic constructions and Herglotz transforms. The main contributions generalize the Korenblum–Roberts results to a broader class of weights and supply a new constructive proof for the El-Fallah–Kellay–Seip-type theorems, including the cyclicity of singular inner functions under finite $oldsymbol{ appa}_W$-entropy and the cyclicity of all zero-free Nevanlinna functions in the divergent-log regime. These methods yield explicit criteria and constructions for cyclicity in growth spaces, with potential applications to Beurling–Carleson-type phenomena and deeper understanding of shift-invariant subspaces in holomorphic function spaces.

Abstract

We revisit the problem of characterizing cyclic elements for the shift operator in a broad class of radial growth spaces of holomorphic functions on the unit disk, focusing on functions of finite Nevanlinna characteristic. We provide results in the range of Dini regular weights, and in the regime of logarithmic integral divergence. Our proofs are largely constructive, enabling us to simplify and extend a classical result by Korenblum and Roberts, and a recent Theorem due to El-Fallah, Kellay, and Seip.

Revisiting cyclic elements in growth spaces

TL;DR

The paper characterizes cyclic Nevanlinna functions for the shift on radial growth spaces with finite Nevanlinna characteristic, treating both the slow (Dini-regular) and fast (logarithmic divergence) regimes for good weights. It reduces cyclicity to the singular inner factor, then develops a Roberts-type decomposition adapted to and provides a constructive, corona-free approach to realize cyclicity via explicit bounded holomorphic constructions and Herglotz transforms. The main contributions generalize the Korenblum–Roberts results to a broader class of weights and supply a new constructive proof for the El-Fallah–Kellay–Seip-type theorems, including the cyclicity of singular inner functions under finite -entropy and the cyclicity of all zero-free Nevanlinna functions in the divergent-log regime. These methods yield explicit criteria and constructions for cyclicity in growth spaces, with potential applications to Beurling–Carleson-type phenomena and deeper understanding of shift-invariant subspaces in holomorphic function spaces.

Abstract

We revisit the problem of characterizing cyclic elements for the shift operator in a broad class of radial growth spaces of holomorphic functions on the unit disk, focusing on functions of finite Nevanlinna characteristic. We provide results in the range of Dini regular weights, and in the regime of logarithmic integral divergence. Our proofs are largely constructive, enabling us to simplify and extend a classical result by Korenblum and Roberts, and a recent Theorem due to El-Fallah, Kellay, and Seip.

Paper Structure

This paper contains 17 sections, 17 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Cyclic Nevanlinna functions in growth spaces
  3. Dini-regular weights
  4. Logarithmic integral divergence
  5. Notations and organization
  6. General properties of $A^p(W)$
  7. Compact embeddings
  8. Cyclic elements in $A^p(W)$
  9. Dini-regular weights
  10. Cyclic inner functions
  11. A Roberts decomposition
  12. The main construction
  13. Logarithmic integral divergence
  14. Cyclicity of inner functions
  15. Reformulating the logarithmic integral divergence
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $0<p \leq \infty$ and $W$ be a good weight which satisfies the condition EQ:LogDiniReg. Then a function $f\in A^p(W) \cap \mathcal{N}$ is cyclic in $A^p(W)$ if $\sigma_f(K) = 0$ for all sets $K\subset \partial\mathbb{D}$ of finite $\kappa_W$-entropy.

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • ...and 21 more