Dirac structures on Hilbert spaces

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 〉 ) . 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 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.