Dirac structures on Hilbert spaces

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〉). 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.

Keywords

• Dirac structure
• maximally isotropic
• Hilbert space
• L-admissibility.