Let $(X,\mu)$ be a measured space with probability measure $\mu$ and $Aut(X,\mu)$ denote the group of measure preserving bijections of $(X,\mu)$. It is easy to see that there is a homomorphism from $Aut(X,\mu)$ to the group of unitary operators on $L^2(X,\mu)$, denoted by $\mathcal{U}(L^2(X,\mu))$. I want to know some natural examples of unitary operators in $\mathcal{U}(L^2(X,\mu))$ which do not come from $Aut(X,\mu)$.

Thank you in advance!

P.S. I am already aware of the following type of elementary constructions: Suppose $X$ is a finite set with uniform probability measure and $U$ is a $n×n$ unitary matrix which is not a permutation matrix.

I am interested in the examples where $(X,\mu)$ is measured isomorphic to $[0,1]$ with Lebesgue measure.

$\DeclareMathOperator\Aut{Aut}$Let $(X,\mu)$ be a measured space with probability measure $\mu$ and $\Aut(X,\mu)$ denote the group of measure preserving bijections of $(X,\mu)$. It is easy to see that there is a homomorphism from $\Aut(X,\mu)$ to the group of unitary operators on $L^2(X,\mu)$, denoted by $\mathcal{U}(L^2(X,\mu))$. I want to know some natural examples of unitary operators in $\mathcal{U}(L^2(X,\mu))$ which do not come from $\Aut(X,\mu)$.

P.S. I am already aware of the following type of elementary constructions: Suppose $X$ is a finite set with uniform probability measure and $U$ is a $n\times n$ unitary matrix which is not a permutation matrix.

I am interested in the examples where $(X,\mu)$ is measured isomorphic to $[0,1]$ with Lebesgue measure.