Symplectic forms on Banach spaces

TL;DR

This paper explores symplectic structures on Banach spaces, extending previous results to higher order Rochgberg spaces and introducing almost symplectic structures, revealing new geometric properties and structures.

Contribution

It generalizes the existence of symplectic structures without Lagrangian subspaces to higher order Rochgberg spaces and introduces almost symplectic structures on Banach spaces.

Findings

Higher order Rochgberg spaces are symplectic with no Lagrangian subspaces.

Natural duality induces symplectic structures on even spaces.

Complex structures are needed for odd spaces' symplectic structures.

Abstract

We extend and generalize the result of Kalton and Swanson ($Z_2$ is a symplectic Banach space with no Lagrangian subspace) by showing that all higher order Rochgberg spaces $\mathfrak R^{(n)}$ are symplectic Banach spaces with no Lagrangian subspaces. The nontrivial symplectic structure on even spaces is the one induced by the natural duality; while the nontrivial symplectic structure on odd spaces requires perturbation with a complex structure. We will also study symplectic structures on general Banach spaces and, motivated by the unexpected appearance of complex structures, we introduce and study almost symplectic structures.