LAUSR

Let $$C_{\varphi }$$Cφ be the composition operator with monomial symbol $$\varphi (z)=z^m$$φ(z)=zm, $$z\in \mathbb {D}$$z∈D, for some positive integer m. In this article, we investigate the point spectrum, spectrum, and essential spectrum of the operators $$C_{\varphi }^*C_{\varphi }$$Cφ∗Cφ, $$C_{\varphi }C_{\varphi }^*$$CφCφ∗, self-commutator $$[C_{\varphi }^*,C_{\varphi }]$$[Cφ∗,Cφ] and anti-self-commutator $$\{C_{\varphi }^*,C_{\varphi }\}$${Cφ∗,Cφ} on weighted Hardy spaces $$H^2(\beta )$$H2(β) and recover known results for the classical Hardy, Bergman, and Dirichlet spaces.