Dirac structures on Hilbert spaces
type: Item
instance of: scholarly article
title: Dirac structures on Hilbert spaces
zbMATH Open document ID: 0934.46018
DOI: 10.1155/S0161171299220972
author: K. Appert
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2000-04-25T00:00:00Z
full work available at URL: https://eudml.org/doc/48377
review text: 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.
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46C05
zbMATH DE Number: 1275563
zbMATH Keywords: \(L\)-admissibility
zbMATH Keywords: Dirac structure
zbMATH Keywords: maximally isotropic
reviewed by: Liu Zheng
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1275563