Opuscula Math. 43, no. 3 (2023), 275-333

Opuscula Mathematica

Operators induced by certain hypercomplex systems

Daniel Alpay
Ilwoo Cho

Abstract. In this paper, we consider a family \(\{ \mathbb{H}_{t}\}_{t\in\mathbb{R}}\) of rings of hypercomplex numbers, indexed by the real numbers, which contain both the quaternions and the split-quaternions. We consider natural Hilbert-space representations \(\{(\mathbb{C}^{2},\pi_{t})\}_{t\in\mathbb{R}}\) of the hypercomplex system \(\{ \mathbb{H}_{t}\}_{t\in\mathbb{R}}\), and study the realizations \(\pi_{t}(h)\) of hypercomplex numbers \(h \in \mathbb{H}_{t}\), as \((2\times 2)\)-matrices acting on \(\mathbb{C}^{2}\), for an arbitrarily fixed scale \(t\in\mathbb{R}\). Algebraic, operator-theoretic, spectral-analytic, and free-probabilistic properties of them are considered.

Keywords: scaled hypercomplex ring, scaled hypercomplex monoids, representations, scaled-spectral forms, scaled-spectralization.

Mathematics Subject Classification: 20G20, 46S10, 47S10.

Full text (pdf)

  1. D. Alpay, M.E. Luna-Elizarrarás, M. Shapiro, D. Struppa, Gleason's problem, rational functions and spaces of left-regular functions: The split-quaternion setting, Isr. J. Math. 226 (2018), 319-349. https://doi.org/10.1007/s11856-018-1696-y
  2. I. Cho, P.E.T. Jorgensen, Spectral analysis of equations over quaternions, Conference Proceeding for International Conference on Stochastic Processes and Algebraic Structures from Theory towards Applications (SPAS 2019), Vastras, Sweden (2021).
  3. I. Cho, P.E.T. Jorgensen, Multi-variable quaternionic spectral analysis, Opuscula Math. 41(2021), no. 3, 335-379. https://doi.org/10.7494/OpMath.2021.41.3.335
  4. C. Doran, A. Lasenby, Geometric Algebra for Physicists, Cambridge University Press, Cambridge, 2003
  5. F.O. Farid, Q.-W. Wang, F. Zhang, On the eigenvalues of quaternion matrices, Linear Multilinear Algebra 59 (2011), no. 4, 451-473. https://doi.org/10.1080/03081081003739204
  6. C. Flaut, Eigenvalues and eigenvectors for the quaternion matrices of degree two, An. Ştiinţ. Univ. Ovidius Constanţa Ser. Mat. 10 (2002), no. 2, 39-44.
  7. P.R. Girard, Einstein's equations and Clifford algebra, Adv. Appl. Clifford Algebras 9 (1999), no. 2, 225-230. https://doi.org/10.1007/BF03042377
  8. P.R. Halmos, A Hilbert Space Problem Book, Graduate Texts in Mathematics, vol. 19, Springer-Verlag, New York-Berlin, 1982.
  9. P.R. Halmos, Linear Algebra Problem Book, The Dolciani Mathematical Expositions, vol. 16, Mathematical Association of America, Washington, DC, 1995.
  10. W.R. Hamilton, Lectures on Quaternions, Cambridge Univ. Press., 1853.
  11. I.L. Kantor, A.S. Solodnikov, Hypercomplex Numbers: An Elementary Introduction to Algebras, Springer, Berlin, 1989.
  12. V. Kravchenko, Applied Quaternionic Analysis, Research and Exposition in Mathematics, vol. 28, Heldermann Verlag, Lemgo, 2003.
  13. S.D. Leo, G. Scolarici, L. Solombrino, Quaternionic eigenvalue problem, J. Math. Phys. 43 (2002), no. 11, 5815-5829.
  14. T.S. Li, Eigenvalues and eigenvectors of quaternion matrices, J. Central China Normal Univ. Natur. Sci. 29 (1995), no. 4, 407-411.
  15. N. Mackey, Hamilton and Jacobi meet again: quaternions and the eigenvalue problem, SIAM J. Matrix Anal. Appl. 16 (1995), no. 2, 421-435.
  16. S. Qaisar, L. Zou, Distribution for the standard eigenvalues of quaternion matrices, Int. Math. Forum 7 (2012), no. 17-20, 831-838.
  17. L. Rodman, Topics in Quaternion Linear Algebra, Princeton University Press, Princeton, NJ, 2014.
  18. B.A. Rozenfeld, A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space, Studies in the History of Mathematics and Physical Sciences, vol. 12, Springer, 1988.
  19. R. Speicher, Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, Mem. Amer. Math. Soc. 132 (1998), no. 627. https://doi.org/http://dx.doi.org/10.1090/memo/0627
  20. A. Sudbery, Quaternionic analysis, Math. Proc. Cambridge Philos. Soc. 85 (1979), no. 2, 199-224. https://doi.org/10.1017/S0305004100055638
  21. J. Vince, Geometric Algebra for Computer Graphics, Springer-Verlag London, Ltd., London, 2008.
  22. D.V. Voiculescu, K.J. Dykema, A. Nica, Free Random Variables, American Mathematical Society, Providence, RI, 1992.
  23. J. Voight, Quaternion Algebra, Graduate Texts in Mathematics, 288. Springer, Cham, 2021.
  • Communicated by P.A. Cojuhari.
  • Received: 2022-10-11.
  • Revised: 2023-04-14.
  • Accepted: 2023-04-17.
  • Published online: 2023-05-17.
Opuscula Mathematica - cover

Cite this article as:
Daniel Alpay, Ilwoo Cho, Operators induced by certain hypercomplex systems, Opuscula Math. 43, no. 3 (2023), 275-333, https://doi.org/10.7494/OpMath.2023.43.3.275

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

We advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.