Bounded Composition Operators on Hilbert Space of Complex-Valued Harmonic Functions

TL;DR

This work addresses the boundedness and isometric behavior of composition operators on the Hilbert space $\mathnormal{H}_{h}^{2}(\mathbb{D})$ of complex-valued harmonic functions. It extends classical results from analytic Hardy spaces to the harmonic setting, leveraging Poisson integrals and reproducing kernels to connect the symbol $\phi$ with operator properties. The authors provide a complete isometry classification for $C_{\phi}$, explicit norm bounds in terms of $|\phi(0)|$, and a kernel-transform relationship $C_{\phi}^{*}K_{\alpha}=K_{\phi(\alpha)}$, offering a framework to analyze spectral and kernel-based aspects in this context. These results clarify how the geometry of self-maps of the disk governs operator behavior on harmonic function spaces and furnish tools for further investigations in harmonic analysis and operator theory.

Abstract

In this paper, we study composition operators on Hilbert space of complex-valued harmonic functions. In particular, we explore isometries, the type of self-map that generate bounded composition operator, and characterize the boundedness of composition operator in terms of Poisson integral. Furthermore, we establish the relation between reproducing kernels and composition operators on Hilbert space of complex-valued harmonic functions.

Theorems & Definitions (23)

• Definition 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Corollary 2.7
• Theorem 3.1
• proof
• Theorem 3.2