Abstract
Let \(\mathcal {H}\) be a right quaternionic Hilbert space and let T be a bounded quaternionic normal operator on \(\mathcal {H}\). In this article, we show that T can be factorized in a strongly irreducible sense, that is, for any \(\delta >0\) there exist a compact operator K with the norm \(\Vert K\Vert < \delta\), a partial isometry W and a strongly irreducible operator S on \(\mathcal {H}\) such that
We illustrate our result with an example. In addition, we discuss the quaternionic version of the Riesz decomposition theorem and obtain a consequence that if the S-spectrum of a bounded (need not be normal) quaternionic operator is disconnected by a pair of disjoint axially symmetric closed subsets, then the operator is strongly reducible.
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Acknowledgements
We thank NBHM (National Board for Higher Mathematics, India) for financial support with ref No. 0204/66/2017/R&D-II/15350, and also Indian Statistical Institute Bangalore for providing necessary facilities to carry out this work.
We wish to express our sincere gratitude to Professor B.V. Rajarama Bhat for useful discussions and suggestions.
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Communicated by Jan Stochel.
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Pamula, S.K. Strongly irreducible factorization of quaternionic operators and Riesz decomposition theorem. Banach J. Math. Anal. 15, 9 (2021). https://doi.org/10.1007/s43037-020-00084-9
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DOI: https://doi.org/10.1007/s43037-020-00084-9
Keywords
- Axially symmetric set
- Quaternionic Hilbert space
- S-specturm
- Strongly irreducible operator
- Riesz decomposition theorem