Symplectic Nonsqueezing in Infinite-Dimensional Hilbert Spaces

Definitions and mathematical statement

The supplied question is: "Does the finite-dimensional symplectic non-squeezing theorem extend to symplectomorphisms of infinite-dimensional Hilbert symplectic spaces?"

The record does not specify the class of symplectic structures, the regularity of the maps, or whether the maps are globally defined. Consequently, it does not determine a unique quantified conjecture. The following is a precise candidate in the strong, compatible Hilbert setting, with these additional choices made explicit.

Let $H$ be an infinite-dimensional real Hilbert space with inner product $\langle\cdot,\cdot\rangle$, and let $J:H\to H$ be a bounded linear operator satisfying \[ J^2=-I,\qquad J^*=-J,\qquad J^*J=I. \] Define the symplectic form $\omega(u,v)=\langle Ju,v\rangle$. For a unit vector $e\in H$, let $P_e$ be the orthogonal projection onto the two-dimensional symplectic plane $\operatorname{span}_{{\mathbb{R}}}\{e,Je\}$. For $r,R>0$, define the open ball and open symplectic cylinder by \[ B_H(r)=\{x\in H:\|x\|<r\},\qquad Z_e(R)=\{x\in H:\|P_ex\|<R\}. \] A smooth symplectomorphism $\Phi:U\to V$ between open subsets of $H$ means a $C^\infty$ diffeomorphism, with smoothness understood in the Fréchet sense, such that \[ \omega\left(D\Phi(x)u,D\Phi(x)v\right)=\omega(u,v) \quad\text{for every }x\in U\text{ and }u,v\in H. \] A candidate non-squeezing assertion asks whether, for every such $(H,J)$, every unit vector $e$, every $r,R>0$, every pair of open sets $U,V\subseteq H$ with $B_H(r)\subseteq U$, and every such symplectomorphism $\Phi:U\to V$, one has \[ \Phi\left(B_H(r)\right)\subseteq Z_e(R) \quad\Longrightarrow\quad r\le R. \] This candidate uses maps defined on a neighborhood of the ball. Restricting instead to global symplectomorphisms $H\to H$ gives another precise version. The supplied record does not identify which version is intended, whether weak symplectic forms are included, or which differentiability or additional analytic hypotheses are allowed. These choices require mathematical review before either version can be identified with the source problem.

Short English statement

The question asks whether symplectic transformations in an infinite-dimensional Hilbert space can place a ball inside a symplectic cylinder of smaller radius. A precise candidate is given for smooth maps and a compatible strong symplectic structure; the source leaves the intended hypotheses unresolved. Its supplied status is partially solved, subject to the dated, unverified assessment below.

Sources

• Local UnsolvedMath registry, permanent ID $30004068$, source problem code catalogue entry, in the supplied Oberwolfach Reports Open Problems record. The record lists proposed year 2019 and names no proposer. It labels the statement "exact" and describes it as publication-ready; these are supplied registry assertions, not independent verification of the formulation.
• Supplied report citation: Nonlinear Evolution Equations: Analysis and Numerics (2020), Oberwolfach Reports, pp. 305--405, DOI 10.4171/owr/2019/5. https://doi.org/10.4171/owr/2019/5. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Abbondandolo--Majer, "convex Hilbert images." https://arxiv.org/abs/1405.3200.
• Supplied literature reference: Sukhov--Tumanov, "Hilbert-space theorem." https://arxiv.org/abs/1411.3989.
• Supplied literature reference: Cicmilović, "infinite-dimensional non-squeezing thesis." https://hdl.handle.net/20.500.11811/9440.
• Supplied literature assessment, checked 2026-08-22: the record labels the problem partially solved and states that infinite-dimensional non-squeezing is proved for convex images, Hilbert-scale regular symplectomorphisms, and several Hamiltonian PDE flows, while the unrestricted Hilbert-space analogue remains unproved. This is supplied dated curation derived from the OWR curation, not independently verified. Its time-sensitive status and the precise hypotheses of the cited results require independent verification.