On symplectic Banach spaces

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 R-(n) are symplectic Banach spaces with no Lagrangian subspaces. The nontrivial symplectic structure on Rochberg spaces of even order is the one induced by the natural duality; while the nontrivial symplectic structure on Rochberg spaces of odd order 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.