Composition Operators with Monomial Symbol Acting on Weighted Hardy Spaces

Abstract

Let $$C_{\varphi }$$ be the composition operator with monomial symbol $$\varphi (z)=z^m$$ , $$z\in \mathbb {D}$$ , for some positive integer m. In this article, we investigate the point spectrum, spectrum, and essential spectrum of the operators $$C_{\varphi }^*C_{\varphi }$$ , $$C_{\varphi }C_{\varphi }^*$$ , self-commutator $$[C_{\varphi }^*,C_{\varphi }]$$ and anti-self-commutator $$\{C_{\varphi }^*,C_{\varphi }\}$$ on weighted Hardy spaces $$H^2(\beta )$$ and recover known results for the classical Hardy, Bergman, and Dirichlet spaces.