Abstract
The origins of an interesting but still little known operator trigonometry as it developed from perturbation questions in the Hille-Yosida-Phillips operator semigroup theory over 30 years ago, will be traced. This operator trigonometry, largely a creation of this author, features a notion of antieigen-value which corresponds to the largest turning angle of an operator. As such, it constitutes an extension of the Rayleigh-Ritz theory, to now include both eigen-vectors and antieigenvectors. Then the application of the operator trigonometry, during the last 10 years, to iterative solvers of linear systems in computational linear algebra, will be described. Roughly, the operator trigonometry provides the previously missing geometrical meanings of these algorithms. For example, the famous Kantorovich bound, known now for 50 years, is now clearly understood geometrically. Finally, a very interesting recent (1998) application to quantum probabilities will be exposed.
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© 2000 Springer Basel AG
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Gustafson, K. (2000). Semigroup Theory and Operator Trigonometry. In: Balakrishnan, A.V. (eds) Semigroups of Operators: Theory and Applications. Progress in Nonlinear Differential Equations and Their Applications, vol 42. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8417-4_12
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DOI: https://doi.org/10.1007/978-3-0348-8417-4_12
Publisher Name: Birkhäuser, Basel
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