- 期刊
DIRAC STRUCTURES ON HILBERT SPACES
摘要
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 = p1(l)+p2(l) for some l ∈ L, where p1 and p2 are projections. When p1(L) is closed, for any Hilbert subspace W ⊂ H, an induced Dirac structure on W is introduced. The latter concept has also been generalized.