Monomial operators

Abstract

We study monomial operators on \(L^2[0, 1]\), that is bounded linear operators that map each monomial \(x^n\) to a multiple of \(x^{pn}\) 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.