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The Finsler geometry of groups of isometries of Hilbert Space

Part of: Lie groups

Published online by Cambridge University Press:  09 April 2009

C. J. Atkin
C. J. Atkin
Affiliation:
Department of Mathematics Victoria University of WellingtonPrivate Bag Wellington, New Zealand
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Abstract

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The paper deals with six groups: the unitary, orthogonal, symplectic, Fredholm unitary, special Fredholm orthogonal, and Fredholm symplectic groups of an infinite-dimensional Hilbert space. When each is furnished with the invariant Finsler structure induced by the operator-norm on the Lie algebra, it is shown that, between any two points of the group, there exists a geodesic realising this distance (often, indeed, a unique geodesic), except in the full orthogonal group, in which there are pairs of points that cannot be joined by minimising geodesics, and also pairs that cannot even be joined by minimising paths. A full description is given of each of these possibilities.

MSC classification

Secondary: 22E65: Infinite-dimensional Lie groups and their Lie algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 42 , Issue 2 , April 1987 , pp. 196 - 222
Copyright
Copyright © Australian Mathematical Society 1987

References

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