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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 $\langle (x,y),(x',y')\rangle +=\frac{1}{2} (\langle x,y'\rangle
+\langle x',y\rangle )$. 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 $z\in H$ is uniquely
decomposed as $z=p_1(l)+p_2(l)$ for some $l\in L$, 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.
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