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N Subspaces

Published online by Cambridge University Press:  20 November 2018

V. S. Sunder
V. S. Sunder*
Affiliation:
Indian Statistical Institute, New Delhi, India
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It is a well-known fact (cf., for instance Lemma 7.3.1 of [8], and also [2] and [4] ) that if M and N are closed subspaces of a finite-dimensional Hilbert space, and if M and N are in ‘generic’ position (i.e., any two of the four subspaces M, M⊥, N, N⊥ have trivial intersection), then N is the graph of a linear isomorphism of M onto M⊥ . To be sure, there exist infinite-dimensional versions of this, where one must allow for unbounded operators in case the ‘gap’ between M and N is zero, in the sense of Kato [7]. (There is an extensive literature on pairs of subspaces, [2], [3], [4], [6] and [7], to cite a few; for a fairly extensive bibliography, see [3].)

This paper addresses itself to the case of n (2 ≦ n < ∞) subspaces.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 40 , Issue 1 , 01 February 1988 , pp. 38 - 54
Copyright
Copyright © Canadian Mathematical Society 1988

References

1. Davis, C., Generators of the ring of bounded operators, Proc. Amer. Math. Soc.. 6 (1955), 970–972.Google Scholar
2. Davis, C., Separation of two linear subspaces, Acta Sci. Math. Szeged. 19 (1958), 172–187.Google Scholar
3. Davis, C. and Kahan, W. M., Rotation of eigenvectors, SIAM J. Numer. Anal. 7 (1970), 1–16.Google Scholar
4. Halmos, P. R., Two subspaces, Trans. Amer. Math. Soc. 144 (1969), 381–389.Google Scholar
5. Hotelling, H., Relations between two sets of variables, Biometrika. 28 (1936), 321–377.Google Scholar
6. Jordan, C., Essai sur la géométrie an dimensions, Bull. Soc. Math. France. 3 (1875), 103–174.Google Scholar
7. Kato, T., Perturbation theory for linear operators (Springer, Berlin, 1966).Google Scholar
8. Murray, F.J. and von Neumann, J., On rings of operators, Ann. of Math. 37 (1936), 116–229.Google Scholar
9. Roy, S. N., Some aspects of multivariate analysis (Wiley, New York, 1958).Google Scholar