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Paper IPM / Physic / 6496  


Abstract:  
For a real Hilbert space (H,〈,〉), a subspace L ⊂ H ⊕H 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 onetoone 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 z ∈ H is uniquely
decomposed as z=p_{1}(l)+p_{2}(l) for some l ∈ L, where p_{1} and p_{2} are
projections. When p_{1} (L) is closed, for any Hilbert subspace W ⊂ H,
an induced Dirac structure on W is introduced. The latter concept has also
been generalized.
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