Let $(M, \omega)$ be a $2n$-dimensional symplectic manifold and let $L_2(M,|\omega^n|)$ be the Hilbert space of complex-valued functions on $M$ that are square integrable with respect to the Liouville measure.

A classical (but still wonderful) remark due to Koopman is that given a measure preserving map $\phi: M \rightarrow M$, the operator $$ U_\phi : L_2(M,|\omega^n|) \longrightarrow L_2(M,|\omega^n|) $$ defined by $f \mapsto f \circ \phi$ is unitary. The operator $U_\phi$ is sometimes called the Koopman operator of the map $\phi$. Since symplectomorphisms are measure preserving, it seems natural to ask the following

Question. What is the uniform closure in the unitary group of the set of Koopman operators of symplectomorphisms?

In particular, are there Koopman operators of measure preserving maps from $\mathbb{R}^4$ to itself that cannot be well approximated by Koopman operators of symplectomorphisms?

The same problem with the strong and weak topologies may also be interesting, but since the question is whether Gromov's non-squeezing theorem or some other symplectic rigidity result can be seen at the level of the Koopman operators, it makes sense to chose the strongest possible topology.

Let $(M, \omega)$ be a $2n$-dimensional symplectic manifold and let $L_2(M,|\omega^n|)$ be the Hilbert space of complex-valued functions on $M$ that are square integrable with respect to the Liouville measure.

A classical (but still wonderful) remark due to Koopman is that given a measure preserving map $\phi: M \rightarrow M$, the operator $$ U_\phi : L_2(M,|\omega^n|) \longrightarrow L_2(M,|\omega^n|) $$ defined by $f \mapsto f \circ \phi$ is unitary. The operator $U_\phi$ is sometimes called the Koopman operator of the map $\phi$. Since symplectomorphisms are measure preserving, it seems natural to ask the following

Vague question. Is there anything at all particular about Koopman operators of symplectomorphisms?