# Self-commutators of composition operators with monomial symbols on the Bergman space

Authors

1 Department of Mathematics, Shiraz University, Shiraz, Iran.

2 Department of Engineering, Khonj Branch, Islamic Azad University, Khonj, Iran.

Abstract

Let $\varphi(z)=z^m, z \in \mathbb{U}$, for some positive integer $m$, and $C_\varphi$ be the composition operator on the Bergman space $\mathcal{A}^2$ induced by $\varphi$. In this article, we completely determine the point spectrum, spectrum, essential spectrum, and essential norm of the operators $C^*_\varphi C_\varphi, C_\varphi C^*_\varphi$ as well as self-commutator and anti-self-commutators of $C_\varphi$. We also find the eigenfunctions of these operators.

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### History

• Receive Date: 04 September 2014
• Revise Date: 25 January 2015
• Accept Date: 05 February 2015