Composition operators between model and Hardy spaces

Abstract: Let $n\ge 1$ and $\varphi: \mathbb{D}<sup>n\to\mathbb{D}$ be a holomorphic function, where $\mathbb{D}$ denotes the open unit disk of $\mathbb{C}$. Let $Θ: \mathbb{D} \to \mathbb{D}$ be an inner function and $K<sup>p_Θ$, $p&gt;0$, denote the corresponding model space. We obtain characterizations of the compact composition operators $$C_\varphi: K^p_Θ\to H<sup>p(\mathbb{D}<sup>n)$$, $1&lt;p&lt;\infty$, where $H<sup>p(\mathbb{D}<sup>n)$ denotes the Hardy space.

• The paper proves that the composition operator is compact if and only if an integral condition involving the Nevanlinna counting function and the inner function’s defect vanishes as the boundary is approached.
• By applying real interpolation theory, the work demonstrates that compactness in the Hilbert space case (p=2) extends to all p in (1,∞), confirming the p-independence of the results.
• The study further characterizes compactness for one-component inner functions through Clark measures, linking the vanishing singular parts of these measures to operator compactness.

Compactness of Composition Operators Between Model and Hardy Spaces

Introduction

This paper, "Composition operators between model and Hardy spaces" (2604.04054), systematically investigates the structure and compactness criteria of composition operators acting between model spaces $K^p_\Theta$ and Hardy spaces $H^p(\mathbb{D}^n)$. For a holomorphic symbol $\varphi: \mathbb{D}^n \to \mathbb{D}$ and an inner function $\Theta$ on the unit disk, the core question addressed is: for which $\varphi$ is the composition operator $C_\varphi: K^p_\Theta \to H^p(\mathbb{D}^n)$ compact? The analysis employs advanced tools—Nevanlinna counting functions and Clark measures—achieving complete, technically nuanced characterizations in the general polydisk context.

Preliminaries and Notation

Hardy spaces $H^p(\mathbb{D}^n)$ and model spaces $K^p_\Theta$ are central to the discourse. Recall,

• $H^p(\mathbb{D}^n)$ denotes the space of holomorphic functions on the polydisk $\mathbb{D}^n$ that satisfy the standard $H^p(\mathbb{D}^n)$0 growth condition on boundary approach.
• Given an inner function $H^p(\mathbb{D}^n)$1 on $H^p(\mathbb{D}^n)$2, $H^p(\mathbb{D}^n)$3, with $H^p(\mathbb{D}^n)$4 being the classical model space $H^p(\mathbb{D}^n)$5.
• The composition operator $H^p(\mathbb{D}^n)$6 is defined by $H^p(\mathbb{D}^n)$7.

The compactness of $H^p(\mathbb{D}^n)$8 is tied intricately to the analytic and geometric properties of the symbol $H^p(\mathbb{D}^n)$9 and the inner function $\varphi: \mathbb{D}^n \to \mathbb{D}$0.

Main Compactness Criteria

Nevanlinna Counting Function Approach

A central role is played by the Nevanlinna counting function $\varphi: \mathbb{D}^n \to \mathbb{D}$1, which encodes the multiplicity of preimages of $\varphi: \mathbb{D}^n \to \mathbb{D}$2 under $\varphi: \mathbb{D}^n \to \mathbb{D}$3 with a logarithmic weight favoring proximity to the boundary. The following equivalence is established:

• $\varphi: \mathbb{D}^n \to \mathbb{D}$4 is compact if and only if

$\varphi: \mathbb{D}^n \to \mathbb{D}$5

Here $\varphi: \mathbb{D}^n \to \mathbb{D}$6 is the normalized measure on the torus $\varphi: \mathbb{D}^n \to \mathbb{D}$7 and $\varphi: \mathbb{D}^n \to \mathbb{D}$8 approaches the distinguished boundary. This result generalizes classical one-variable theorems and highlights the interplay between the symbolic dynamics of $\varphi: \mathbb{D}^n \to \mathbb{D}$9 and the defectiveness of $\Theta$0 near the boundary.

Compactness Independence of $\Theta$1

By leveraging real interpolation theory, specifically the Cwikel compactness theorem for interpolation spaces, the paper demonstrates that compactness of $\Theta$2 is independent of the value of $\Theta$3 in $\Theta$4. In particular, compactness in the Hilbertian case $\Theta$5 implies compactness for all $\Theta$6, and vice versa. This sharp, general result is nontrivial due to the structural complexities of $\Theta$7 for $\Theta$8.

Clark Measure Criterion for One-Component Inner Functions

The paper provides an alternative and equally complete characterization for one-component inner functions using Clark measures. For such $\Theta$9, $\varphi$0 is compact if and only if the singular parts of all Clark measures vanish on the spectrum of $\varphi$1. That is,

$\varphi$2

where $\varphi$3 is the spectrum, and $\varphi$4 is the Clark measure associated to $\varphi$5. This links operator-theoretic compactness to a spectral property of singular measures arising from the inner function, extending and unifying prior results for the disk.

Technical Contributions

• Stanton’s formula is generalized for $\varphi$6, allowing explicit expressions of $\varphi$7-norms post-composition in terms of the Nevanlinna function.
• Subharmonicity of the Nevanlinna counting function is used to establish necessary integral estimates.
• Detailed kernel estimates for model spaces are utilized in the Clark measure direction, especially when sequences tend to the boundary point spectrum.
• The one-sided compactness result for real interpolation (Cwikel’s theorem) is a backbone for independence-of-$\varphi$8 arguments, enabling extension from Hilbert to Banach settings.

Implications and Further Directions

The compactness characterizations elucidated in this paper rigorously unify perspectives from function theory (via inner functions and Clark measures), operator theory (through compactness and interpolation), and several complex variables (for $\varphi$9). The explicit criteria enable, in principle, the determination of compactness for large classes of composition operators arising in function-theoretic and applied settings, such as control theory and systems analysis where model spaces arise.

The dependence on the symbol $C_\varphi: K^p_\Theta \to H^p(\mathbb{D}^n)$0 solely through the Nevanlinna counting function suggests future investigations into the sharpness of such criteria, possible relaxations for more general domains, and stability under perturbations of $C_\varphi: K^p_\Theta \to H^p(\mathbb{D}^n)$1. Extending the Clark measure criteria to $C_\varphi: K^p_\Theta \to H^p(\mathbb{D}^n)$2 and non one-component inner functions remains a compelling theoretical challenge.

Conclusion

The paper achieves a comprehensive solution to the compactness problem for composition operators between model and Hardy spaces in the polydisk, unifying Nevanlinna-type and Clark measure approaches. The independence from the parameter $C_\varphi: K^p_\Theta \to H^p(\mathbb{D}^n)$3 in the Banach range adds practical scope to operator classification. These results provide deep links among analytic function theory, harmonic analysis, and operator theory, setting a strong foundation for further advances in the spectral analysis of non-self-adjoint operators and functional analysis on complex domains.