Isometry and ‎s‎‎pectrum of ‎c‎‎omposition ‎o‎‎perator on $n$th ‎w‎‎eighted ‎s‎‎pace

Abbasi, Ebrahim; Abbasi, Behnaz; Zeinalabedini Charandabi, Zohreh; Baghmisheh, Ma‎hdi

doi:10.22060/ajmc.2025.24126.1375

Isometry and ‎s‎‎pectrum of ‎c‎‎omposition ‎o‎‎perator on $n$th ‎w‎‎eighted ‎s‎‎pace

Articles in Press

Document Type : Original Article

Authors

Ebrahim Abbasi ¹

Behnaz Abbasi ²

Zohreh Zeinalabedini Charandabi ³

Ma‎hdi Baghmisheh ³

¹ Department of Mathematics‎, ‎Mah‎, ‎C,Islamic Azad University‎, ‎Mahabad‎, ‎Iran‎

² Department of Mathematics‎, ‎Tabriz Branch‎, ‎Islamic Azad University‎, ‎Tabriz‎, ‎Iran

³ Department of Mathematics‎, ‎Ta‎. ‎C.‎‎, ‎Islamic Azad University‎, ‎Tabriz‎, ‎Iran

10.22060/ajmc.2025.24126.1375

Abstract

‎Let $n\in\mathbb{N}$ and $\alpha>0$‎. ‎The $n$th weighted space $\mathcal{W}^n_‎\alpha‎$‎, ‎consists of all analytic functions on $\mathbb{D}$ such that‎
‎$$\|f\|_{\mathcal{W}^n_‎\alpha‎}=\sum_{i=0}^{n-1}|f^{(i)}(0)|+\sup_{z\in\mathbb{D}}(1-|z|^2)^‎\alpha‎|f^{(n)}(z)|<\infty.$$‎
‎In this paper‎, ‎the isometry of the composition operator $C_‎\varphi‎‎: ‎\mathcal{W}^n_‎\alpha‎\to \mathcal{W}^n_‎\alpha‎$ will be investigated‎, ‎and the spectrum of isometric composition operators will be determined‎. ‎Furthermore‎, ‎we establish that for $\alpha>n$‎, ‎the subspace $\mathcal{W}^n_{‎\alpha‎,0}$‎ ‎(defined as the closure of functions satisfying $\lim_{|z|\rightarrow 1}\mu(z)|f^{(n)}(z)|=0$) supports a hypercyclic composition operator $C_\varphi$ and for $0<\alpha\leq n$‎, ‎no such hypercyclic composition operator exists on $\mathcal{W}^n_{‎\alpha‎,0}$‎.‎

Keywords

Composition operator

Hypercyclicity

Isometry

Spectrum

Subjects