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$\DeclareMathOperator\Aut{Aut}$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.

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    $\begingroup$ Is it possible to be more specific about what you're looking for? There are a lot of examples you can cook up. One easy approach if your space is $L^2([0,1])$ is to mess with the Fourier series, e.g. Fourier transform your vector, change the Fourier series somehow (apply a permutation or stick some phases on some of the components or something) and Fourier transform back. $\endgroup$
    – ors
    Commented Jun 6 at 14:21
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    $\begingroup$ Generalizing your elementary construction to the case $X\simeq[0,1]$ (or anything else, really), you can consider multiplication operators $M_\psi: f\mapsto f\psi$, where $\psi:X\to\mathbb{C}$ has $|\psi|=1$ almost everywhere. $\endgroup$
    – Ben Johnsrude
    Commented Jun 6 at 15:39
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    $\begingroup$ I think I don't really understand the questiion. Map an orthonormal basis to another one and you get a unitary operator. Make sure that it doesn't map the constant 1-function to itself (or that it maps some positive function to a non-positive function) and it doesn't stem from a measure preserving transformation. $\endgroup$
    – Jochen Glueck
    Commented Jun 6 at 18:19
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    $\begingroup$ @KajalDas The Fourier series gives you an isometry from $L^2([0,1])$ to $l^2$ (the space of square summable sequences. What I was suggesting was to construct a unitary on $L^2([0,1])$ by mapping to $l^2$, then applying some simple unitary on $l^2$ (for example a diagonal one, which just multiplies each component by a complex number of absolute value $a$) then applying the inverse Fourier transform to this new $l^2$ sequence which gives you a new $L^2([0,1])$ function. The Riesz–Fischer theorem ensures this process gives you a unitary on $L^2([0,1])$. $\endgroup$
    – ors
    Commented Jun 7 at 9:48
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    $\begingroup$ @KajalDas This really should just be a MSE question instead of being asked here. Bilateral shifts on the Fourier series, basically all Fourier multipliers with values in the unit circle, and multiplication by any function in $L^\infty([0, 1], \mathbb{T})$ (which is not a.s. $1$) are all examples. Are these “well-studied” enough for you? $\endgroup$
    – David Gao
    Commented Jun 8 at 13:33

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