Monomial Operators

TL;DR

This paper investigates monomial operators on L^2[0,1], showing they are unitarily equivalent to weighted composition operators on Hardy spaces, and characterizes the sequences p_n for which these operators are bounded.

Contribution

It provides a complete characterization of monomial operators on L^2[0,1], including their unitary equivalence to weighted composition operators and conditions for boundedness.

Findings

All monomial operators are unitarily equivalent to weighted composition operators.

Characterization of sequences p_n that can arise in monomial operators.

Boundedness criteria for operators when p_n is a fixed translation of n.

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 $x^{p_n}$ 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.