Abstract
On various occasions, when working with quaternionic linear spaces, there is a need to restrict them to their complex linear structure, then it becomes essential to understand whether the pre-existing internal product or norm in the quaternionic space will continue to be compatible with the complex structure of the new space obtained. There are other situations in which these types of questions arise, for example, if a linear space is originally complex but it turns out that it also admits the quaternionic structure. The objective of this work is to present the different options to change the linearities of some linear spaces and to analyze what happens with the pre-existing algebraic objects: to understand if they still work or if they induce some others that will be compatible with the new linear structure.
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Abramovich, Y.A., Aliprantis, C.D., Sirotkin, G., Troitsky, V.B.: Some open problems and conjectures associated with the invariant subspace problem. Positivity 9, 273–286 (2005)
Adler, S.L.: Quaternionic Quantum Mechanics and Quantum Fields. Oxford University Press, New York (1995)
Alpay, D., Luna-Elizarrarás, M.E., Shapiro, M.: Normes des extensions quaternionique d’operateurs réls. C. R. Acad. Sci. Ser. I Math. 340, 639–643 (2005)
Alpay, D., Luna-Elizarrarás, M.E., Shapiro, M.: On the norms of quaternionic extensions of real and complex linear mappings. Math. Methods Appl. Sci. 30, 1005–1036 (2007)
Alpay, D., Colombo, F., Sabadini, I.: Slice Hyperholomorphic Schur analysis. Operator Theory: Advances and Applications, vol. 256. Birkhäuser/Springer, Cham (2016)
Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Ann. Math. 37(4), 823–843 (1936)
Colombo, F., Gantner, J.: Quaternionic closed operators, fractional powers and fractional diffusion process. Operator Theory: Advances and Applications, vol. 274. Birkhäuser/Springer, Cham (2019)
Colombo, F., Sabadini, I., Struppa, D.C.: Noncommutative Functional Calculus. Theory and Applications of Slice Hyperholomorphic Functions. Progress in Mathematics, vol. 289. Birkhäuser (2011)
Colombo, F., Gantner, J., Kimsey, D.P.: Spectral Theory on the S-spectrum for quaternionic operators. Operator Theory: Advances and Applications, vol. 270. Birkhäuser/Springer, Cham (2018)
Delanghe, R., Brackx, F.: Hypercomplex function theory and Hilbert modules with reproducing kernel. Proc. Lond. Math. Soc. 3(37), 545–576 (1978)
Emch, G.: Mécanique quantique quaternionienne et relativité. I. Helv. Phys. Acta 36, 739–769 (1963)
Finkelstein, D., Jauch, J.M., Speiser, D.: Quaternion quantum mechanics I. European Center for Nuclear Research, Geneva, Report 59(7) (1959)
Finkelstein, D., Jauch, J.M., Speiser, D.: Quaternion quantum mechanics II. European Center for Nuclear Research, Geneva, Report 59(11) (1959)
Finkelstein, D., Jauch, J.M., Speiser, D.: Quaternion quantum mechanics III. European Center for Nuclear Research, Geneva, Report 59(17) (1959)
Glazman, I.M., Ljubic, J.: Finite-Dimensional Linear Analysis: A Systematic Presentation in Problem Form. MIT Press, Cambridge (1974)
Horwitz, L.P., Biedenharn, L.C.: Quaternion quantum mechanics: second quantization and gauge fields. Ann. Phys. 157(2), 432–488 (1984)
Jefferies, B.: Spectral properties of noncommuting operators. Lecture Notes in Mathematics, vol. 1843. Springer, Berlin (2004)
Luna-Elizarrarás, M.E., Shapiro, M.: Preservation of the norms of linear operators acting on some quaternionic function spaces. Oper. Theory Adv. Appl. 157, 205–220 (2005)
Luna-Elizarrarás, M.E., Shapiro, M.: On some relations between real, complex and quaternionic linear spaces. In: Begehr, H., Nicolosi, F. (eds.) More Progresses in Analysis, pp. 99–1008. World Scientific, Singapore (2009)
Luna-Elizarrarás, M.E., Ramirez-Reyes, F., Shapiro, M.: Complexification of real spaces: general aspects. Georg. Math. J. 19(2), 259–282 (2012)
Qian, T., Li, P.: Singular Integrals and Fourier Theory on Lipschitz Boundaries. Science Press, Beijing; Springer, Singapore (2019)
Rastall, P.: Quaternions in relativity. Rev. Mod. Phys. 36, 820–832 (1964)
Razon, A., Horwitz, L.P., Biedenharn, L.C.: On a basic theorem of quaternion modules. J. Math. Phys. 30(1), 59 (1989)
Rembielinski, J.: Quaternionic Hilbert space and colour confinement: I. J. Phys. A Math. Gen. 13, 15–22 (1980)
Rembielinski, J.: Quaternionic Hilbert space and colour confinement: II. The admissible symmetry groups. J. Phys. A Math. Gen. 13, 23–30 (1980)
Sharma, C.S., Almeida, D.F.: Additive functionals and operators on a quaternionic Hilbert space. J. Math. Phys. 30(2), 369–375 (1989)
Lomonosov, V.I.: Invariant subspaces for the family of operators which commute with a completely continuous operator. Funct. Anal. Appl. 7(3), 213–214 (1974). https://doi.org/10.1007/BF01080698
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This article is part of the Topical Collection on Proceedings ICCA 12, Hefei, 2020, edited by Guangbin Ren, Uwe Kähler, Rafał Abłamowicz, Fabrizio Colombo, Pierre Dechant, Jacques Helmstetter, G. Stacey Staples, Wei Wang.
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Luna-Elizarrarás, M.E. On Interactions of Quaternionic and Complex Structures of Linear Spaces. Adv. Appl. Clifford Algebras 31, 60 (2021). https://doi.org/10.1007/s00006-021-01156-1
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DOI: https://doi.org/10.1007/s00006-021-01156-1