International Journal of Mathematics and Mathematical Sciences
Volume 22 (1999), Issue 1, Pages 97-108

Dirac structures on Hilbert spaces

A. Parsian1 and A. Shafei Deh Abad2

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 (H,,), a subspace LHH is said to be a Dirac structure on H if it is maximally isotropic with respect to the pairing (x,y),(x,y)+=(1/2)(x,y+x,y). By investigating some basic properties of these structures, it is shown that Dirac structures on H are in one-to-one correspondence with isometries on H, and, any two Dirac structures are isometric. It is, also, proved that any Dirac structure on a smooth manifold in the sense of [1] 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 L on H, every zH is uniquely decomposed as z=p1(l)+p2(l) for some lL, where p1 and p2 are projections. When p1(L) is closed, for any Hilbert subspace WH, an induced Dirac structure on W is introduced. The latter concept has also been generalized.