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Real-linear operators on quaternionic Hilbert space


Author: N. C. Powers
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
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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 [Enhancements On Off] (What's this?)

  • [1] 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.
  • [2] O. Teichmüller, Operatoren im Wachsschen Raum, J. Reine Angew. Math. 174 (1935), 73-124.
  • [3] K. Viswanath, Normal operators on quaternionic Hilbert space, Trans. Amer. Math. Soc. 162 (1971), 337-350. MR 44 #2067. MR 0284843 (44:2067)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0318965-9
Keywords: Quaternions, quaternionic Hilbert space, quaternionic matrix, real-linear operator, symplectic image
Article copyright: © Copyright 1973 American Mathematical Society

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