International Journal of Mathematics and Mathematical Sciences
Volume 22 (1999), Issue 1, Pages 97-108
Dirac structures on Hilbert spaces
1Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-1745, Tehran, Iran
2Department of Mathematics, University of Tehran, P.O. Box 14155-6455, Tehran, Iran
Received 5 May 1997; Revised 4 August 1997
Copyright © 1999 A. Parsian and A. Shafei Deh Abad. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
For a real Hilbert space , a subspace is said to be a Dirac structure on if it is maximally isotropic with respect to the pairing . By investigating some basic properties of these structures, it is
shown that Dirac structures on are in one-to-one correspondence with isometries on , and, any two Dirac structures are isometric. It is, also, proved that any Dirac structure on a smooth manifold in the sense of  yields a Dirac structure on some Hilbert space. The graph of any densely defined skew symmetric linear operator on a Hilbert space is, also, shown to be a Dirac structure. For a Dirac structure on , every is uniquely decomposed as for some , where and are projections. When is closed, for any Hilbert subspace , an induced Dirac structure on is introduced. The latter concept has also been generalized.