Dirac structures on Hilbert spaces
For a real Hilbert space \((H,\langle,\rangle)\), a subspace \(L\subset H\oplus H\) is said to be a Dirac structure on \(H\) if it is maximally isotropic with respect to the pairing \(\langle(x,y),(x',y')\rangle_+= (1/2)(\langle x,y'\rangle+\langle x',y\rangle)\). 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 \textit{T. J. Courant} [Trans. Am. Math. Soc. 319, No. 2, 631-661 (1990; Zbl 0850.70212)] 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 \(z\in H\) is uniquely decomposed as \(z= p_1(\ell)+ p_2(\ell)\) for some \(\ell\in L\), where \(p_1\) and \(p_2\) are projections. When \(p_1(L)\) is closed, for any Hilbert subspace \(W\subset H\), an induced Dirac structure on \(W\) is introduced. The latter concept has also been generalized.
scientific article; zbMATH DE number 1275563
scientific article