Hilbert space idempotents and involutions
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- by Don Buckholtz PDF
- Proc. Amer. Math. Soc. 128 (2000), 1415-1418 Request permission
Abstract:
Norms of idempotents, involutions, and the Hermitian and skew-Hermitian parts of involutions are shown to be elementary trigonometric functions of an angle between two subspaces of Hilbert space. When the spaces involved are nontrivial, the norm of a linear idempotent is the cosecant of the angle between its range and kernel; the norm of a linear involution is the cotangent of half the angle between the involution’s eigenspaces.References
- Don Buckholtz, Inverting the difference of Hilbert space projections, Amer. Math. Monthly 104 (1997), no. 1, 60–61. MR 1426419, DOI 10.2307/2974825
- Frank Deutsch, von Neumann’s alternating method: the rate of convergence, Approximation theory, IV (College Station, Tex., 1983) Academic Press, New York, 1983, pp. 427–434. MR 754371
- Paul Richard Halmos, A Hilbert space problem book, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 17, Springer-Verlag, New York-Berlin, 1982. MR 675952, DOI 10.1007/978-1-4684-9330-6
Additional Information
- Don Buckholtz
- Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
- Email: mat236@ukcc.uky.edu
- Received by editor(s): October 25, 1996
- Received by editor(s) in revised form: July 2, 1998
- Published electronically: October 5, 1999
- Communicated by: Palle E. T. Jorgensen
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 1415-1418
- MSC (1991): Primary 46C05; Secondary 47A05, 47A30
- DOI: https://doi.org/10.1090/S0002-9939-99-05233-8
- MathSciNet review: 1653425