A symplectic Banach space with no Lagrangian subspaces

Authors:
N. J. Kalton and R. C. Swanson

Journal:
Trans. Amer. Math. Soc. **273** (1982), 385-392

MSC:
Primary 58B20; Secondary 46B99

MathSciNet review:
664050

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we construct a symplectic Banach space 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 is settled in the negative. The proof also shows that admits a nontrivial continuous homomorphism into where is a Hilbert space.

**[1]**Frank F. Bonsall and John Duncan,*Complete normed algebras*, Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80. MR**0423029****[2]**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****[3]**J. J. Duistermaat,*On the Morse index in variational calculus*, Advances in Math.**21**(1976), no. 2, 173–195. MR**0649277****[4]**Richard H. Herman,*On the uniqueness of the ideals of compact and strictly singular operators*, Studia Math.**29**(1967/1968), 161–165. MR**0222657****[5]**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**, 10.1007/BF02762873**[6]**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**, 10.1090/S0002-9947-1979-0542869-X**[7]**E. C. Lance,*Quadratic forms on Banach spaces*, Proc. London Math. Soc. (3)**25**(1972), 341–357. MR**0308742****[8]**R. C. Swanson,*Linear symplectic structures on Banach spaces*, Rocky Mountain J. Math.**10**(1980), no. 2, 305–317. MR**575305**, 10.1216/RMJ-1980-10-2-305**[9]**R. C. Swanson,*Fredholm intersection theory and elliptic boundary deformation problems. I*, J. Differential Equations**28**(1978), no. 2, 189–201. MR**491049**, 10.1016/0022-0396(78)90066-9**[10]**Alan Weinstein,*Symplectic manifolds and their Lagrangian submanifolds*, Advances in Math.**6**(1971), 329–346 (1971). MR**0286137**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
58B20,
46B99

Retrieve articles in all journals with MSC: 58B20, 46B99

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1982-0664050-6

Keywords:
Symplectic Banach space,
isotropic,
Lagrangian subspace,
strictly singular operators,
-algebras

Article copyright:
© Copyright 1982
American Mathematical Society