Real-linear operators on quaternionic Hilbert space
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- by N. C. Powers PDF
- Proc. Amer. Math. Soc. 40 (1973), 1-8 Request permission
Abstract:
The main result is that any continuous real-linear operator $A$ on a quaternionic Hilbert space has a unique decomposition $A = {A_0} + {i_1}{A_1} + {i_2}{A_2} + {i_3}{A_3}$, where the ${A_\nu }$ are continuous linear operators and $({i_{1,}},{i_2},{i_3})$ is any right-handed orthonormal triad of vector quaternions. Other results concern the place of the colinear and complex-linear operators in this characterisation and the effect on the ${A_\nu }$ of a rotation of the triad of vector quaternions. A new result concerning symplectic images of a quaternionic Hilbert space is also presented.References
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J. M. Jauch, Projective representations of the Poincaré group in a quaternionic Hilbert space, Group Theory and its Applications, E. Loebl (editor), Academic Press, New York, 1968, pp. 131-182.
O. Teichmüller, Operatoren im Wachsschen Raum, J. Reine Angew. Math. 174 (1935), 73-124.
- K. Viswanath, Normal operations on quaternionic Hilbert spaces, Trans. Amer. Math. Soc. 162 (1971), 337–350. MR 284843, DOI 10.1090/S0002-9947-1971-0284843-X
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 40 (1973), 1-8
- MSC: Primary 47B99; Secondary 46C05
- DOI: https://doi.org/10.1090/S0002-9939-1973-0318965-9
- MathSciNet review: 0318965