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“School of Physic”

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Paper   IPM / Physic / 6496
School of Physics
  Title:   Dirac Structures on Hilbert Spaces
1.  A. Parsian
2.  A. Shafei Deh Abad
  Status:   Published
  Journal: Int. J. Math. Math. Sci.
  No.:  84
  Vol.:  22
  Year:  1999
  Pages:   97-108
  Supported by:  IPM
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〉). 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.

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