Spectrum of weighted composition operators. Part XI. The essential spectra of some weighted composition operators on the disc algebra

TL;DR

This work analyzes the essential and semi-Fredholm spectra of weighted composition operators on the disc algebra $\mathbb{A}$, focusing on operators of the form $(Tf)(z) = w(z) f(B(z))$ where $B$ is a finite Blaschke product of degree $d \ge 2$ that is elliptic or double parabolic. The authors combine operator-theoretic methods with complex-dynamics tools, leveraging results on periodic ergodic measures for $\varphi(z) = z^d$ and open-map dynamics to derive spectral descriptions. They prove that when $w$ has no zeros on the unit circle, the spectral radius $\rho(T) > 0$ and $\sigma(T) = \sigma_{lsf}(T) = \rho(T) \mathbb{D}$ with $\sigma_{usf}(T) = \rho(T) \mathbb{T}$, with rotation-invariant spectra; when $w$ vanishes on the circle, they obtain partial results (Theorems $t4$, $t5$, $t6$, $t11$) and construct cases where $\sigma_{usf}(T)$ can realize unions $\bigcup_j \lambda_j \mathbb{T}$ or explicit radii depending on parity. These findings illustrate a deep connection between spectral properties and the underlying dynamical system, and suggest further conjectures (including a potential converse in the elliptic/double-parabolic regime).

Abstract

We obtain a complete description of semi-Fredholm spectra of operators of the form $(Tf)(z) = w(z)f(B(z)$ acting on the disc algebra in the case when $B$ is either elliptic or double parabolic finite Blaschke product of degree $d \geq 2$ and $w$ has no zeros on the unit circle. In the case when $B$ has zeros on the unit circle we provide only some partial results. Our results hint on the possibility of interesting connections between the spectral properties of weighted composition operators and complex dynamics.

Theorems & Definitions (28)

• Theorem 2.1
• Theorem 2.2
• Definition 2.3
• Theorem 2.4
• Theorem 2.5
• Lemma 2.6
• proof
• Example 2.7
• Corollary 2.8
• Lemma 2.9