Weighted Composition Operators on Weighted Hardy Spaces

Abstract

For functions $$ \mu : {\mathbb {N}}_0 \rightarrow {\mathbb {C}} $$ and $$ \theta : {\mathbb {N}}_0 \rightarrow {\mathbb {N}}_0 $$ , the weighted composition operator is the operator $$ \varGamma _{\mu , \theta }: {\mathcal {H}}_\beta ^p \rightarrow {\mathbb {C}}^{{\mathbb {N}}_0}$$ defined by $$ \varGamma _{\mu , \theta }(x) = \bigl ( \sum _{m=0}^k\mu (k-m) x_{\theta (m)}\bigr )_{k =0}^\infty $$ for all $$ x = (x_k) \in {\mathcal {H}}_\beta ^p $$ . This paper deals with the properties of weighted composition operators on weighted Hardy spaces $$ {\mathcal {H}}_\beta ^p$$ for $$ 1 \le p \le \infty $$ . The necessary and sufficient conditions are investigated for a weighted composition operator to be compact. A compact subset of non-negative real numbers containing the essential norm of $$ \varGamma _{\mu , \theta } $$ on $$ {\mathcal {H}}_\beta ^p$$ is also computed. The condition under which weighted composition operators commute is also explored.