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

‎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.