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On bounded skew-symmetric forms

Published online by Cambridge University Press:  20 January 2009

Aurel Wintner
Aurel Wintner
Affiliation:
The Johns Hopkins University.
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It is well known that the real, skew-symmetric, non-singular, bilinear forms of n + n variables have no invariants. In fact, each of these forms may be transformed into one and the same form, for instance into the one which occurs in the usual representation of the complex group. The standard proofs of this theorem break down in case of infinite forms which are bounded in the sense of Hilbert, one of the impediments being the possibility of a continuous spectrum. The object of this note is to show that, while the usual proofs break down, the theorem itself is true in Hubert's case also.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 5 , Issue 2 , June 1937 , pp. 90 - 92
Copyright
Copyright © Edinburgh Mathematical Society 1937

References

page 90 note 1 Simple examples of real, bounded, non-singular, skew-symmetric forms with continuous spectra may be deduced, according to Toeplitz, from his theory of L-forms. CfHellinger, B. and Toeplitz, O., Encyklopadie der mathematischen Wissenschaften, II C 13, §44.Google Scholar

page 90 note 2 Wintner, A., “On non-singular bounded matrices,” American Journal of Mathematics, 54 (1932), 145–149.CrossRefGoogle ScholarCfHellinger, E. and Toeplitz, O., loc. cit., footnote 522a.Google Scholar

page 91 note 1 CfWintner, A., loc. cit.Google Scholar

page 91 note 2 CfHellinger, E. and Toeplifcz, O., loc. cit., §41.Google Scholar In the proof given there it is assumed that the matrix is completely continuous (vollstetig). Actually, the proof applies without change in all cases where there is no continuous spectrum.