Abstract and Applied Analysis
Volume 2011 (2011), Article ID 618968, 9 pages
http://dx.doi.org/10.1155/2011/618968
Research Article

Self-Commutators of Composition Operators with Monomial Symbols on the Dirichlet Space

A. Abdollahi and S. Mehrangiz

Department of Mathematics, College of Sciences, Shiraz University, Shiraz 71454, Iran

Received 14 June 2011; Accepted 10 August 2011

Academic Editor: Detlev Buchholz

Copyright © 2011 A. Abdollahi and S. Mehrangiz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let 𝜑 ( 𝑧 ) = 𝑧 𝑛 , 𝑧 ∈ 𝕌 , for some positive integer 𝑛 and 𝐶 𝜑 the composition operator on the Dirichlet space 𝒟 induced by 𝜑 . In this paper, we completely determine the point spectrum, spectrum, essential spectrum, and essential norm of the operators 𝐶 ∗ 𝜑 𝐶 𝜑 , 𝐶 𝜑 𝐶 ∗ 𝜑 and self-commutators of 𝐶 𝜑 , which expose that the spectrum and point spectrum coincide. We also find the eigenfunctions of the operators.