A non-squeezing theorem for convex symplectic images of the Hilbert ball

TL;DR

This paper extends Gromov's non-squeezing theorem to infinite-dimensional symplectic Hilbert spaces for convex images, introducing a new symplectic capacity and exploring unique infinite-dimensional behaviors.

Contribution

It establishes a non-squeezing result in infinite dimensions for convex symplectic images and constructs a symplectic capacity using duality methods.

Findings

Non-squeezing theorem holds in infinite-dimensional convex setting

Constructs a symplectic capacity for convex neighborhoods

Identifies behaviors unique to infinite-dimensional symplectic spaces

Abstract

We prove that the non-squeezing theorem of Gromov holds for symplectomorphisms on an infinite-dimensional symplectic Hilbert space, under the assumption that the image of the ball is convex. The proof is based on the construction by duality methods of a symplectic capacity for bounded convex neighbourhoods of the origin. We also discuss some examples of symplectomorphisms on infinite-dimensional spaces exhibiting behaviours which would be impossible in finite dimensions.