Monomial Operators

We study monomial operators on $L^2[0,1]$, that is bounded linear operators that map each monomial $x^n$ to a multiple of $x^{p_n}$ for some $p_n$. We show that they are all unitarily equivalent to weighted composition operators on a Hardy space. We characterize what sequences $p_n$ can arise. In the case that $p_n$ is a fixed translation of $n$, we give a criterion for boundedness of the operator.