A symplectic Banach space with no Lagrangian subspaces
HTML articles powered by AMS MathViewer
- by N. J. Kalton and R. C. Swanson PDF
- Trans. Amer. Math. Soc. 273 (1982), 385-392 Request permission
Abstract:
In this paper we construct a symplectic Banach space $(X,\Omega )$ which does not split as a direct sum of closed isotropic subspaces. Thus, the question of whether every symplectic Banach space is isomorphic to one of the canonical form $Y \times {Y^ \ast }$ is settled in the negative. The proof also shows that $\mathfrak {L}(X)$ admits a nontrivial continuous homomorphism into $\mathfrak {L}(H)$ where $H$ is a Hilbert space.References
- Frank F. Bonsall and John Duncan, Complete normed algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80, Springer-Verlag, New York-Heidelberg, 1973. MR 0423029
- Paul R. Chernoff and Jerrold E. Marsden, Properties of infinite dimensional Hamiltonian systems, Lecture Notes in Mathematics, Vol. 425, Springer-Verlag, Berlin-New York, 1974. MR 0650113
- J. J. Duistermaat, On the Morse index in variational calculus, Advances in Math. 21 (1976), no. 2, 173–195. MR 649277, DOI 10.1016/0001-8708(76)90074-8
- Richard H. Herman, On the uniqueness of the ideals of compact and strictly singular operators, Studia Math. 29 (1967/68), 161–165. MR 222657, DOI 10.4064/sm-29-2-161-165
- W. B. Johnson, J. Lindenstrauss, and G. Schechtman, On the relation between several notions of unconditional structure, Israel J. Math. 37 (1980), no. 1-2, 120–129. MR 599307, DOI 10.1007/BF02762873
- N. J. Kalton and N. T. Peck, Twisted sums of sequence spaces and the three space problem, Trans. Amer. Math. Soc. 255 (1979), 1–30. MR 542869, DOI 10.1090/S0002-9947-1979-0542869-X
- E. C. Lance, Quadratic forms on Banach spaces, Proc. London Math. Soc. (3) 25 (1972), 341–357. MR 308742, DOI 10.1112/plms/s3-25.2.341
- R. C. Swanson, Linear symplectic structures on Banach spaces, Rocky Mountain J. Math. 10 (1980), no. 2, 305–317. MR 575305, DOI 10.1216/RMJ-1980-10-2-305
- R. C. Swanson, Fredholm intersection theory and elliptic boundary deformation problems. I, J. Differential Equations 28 (1978), no. 2, 189–201. MR 491049, DOI 10.1016/0022-0396(78)90066-9
- Alan Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Advances in Math. 6 (1971), 329–346 (1971). MR 286137, DOI 10.1016/0001-8708(71)90020-X
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 273 (1982), 385-392
- MSC: Primary 58B20; Secondary 46B99
- DOI: https://doi.org/10.1090/S0002-9947-1982-0664050-6
- MathSciNet review: 664050