Symplectic forms on Banach spaces

2022/04/01 by Jesús M. F. Castillo, Castillo, Jesús M. F., Wilson Cuellar +5
Mathematics · #46B10 #46B20 #46B70 #46M18 #Advanced Algebra and Geometry #Advanced Mathematical Physics Problems #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA)

paper · pdf · doi:10.48550/arxiv.2204.01703

openalex publication_date 2022/04/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We extend and generalize the result of Kalton and Swanson (Z₂ 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.

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