Skip to main content
SpringerLink
Log in

Unbounded Normal Operators in Octonion Hilbert Spaces and Their Spectra

  • Published:
Advances in Applied Clifford Algebras Aims and scope Submit manuscript

Abstract

Affiliated and normal operators in octonion Hilbert spaces are studied. Theorems about their properties and of related algebras are demonstrated. Spectra of unbounded normal operators are investigated.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J.C. Baez, The octonions. Bull. Amer. Mathem. Soc. 39: 2 (2002), 145-205.

  2. F. Brackx, R. Delanghe, F. Sommen,Clifford analysis. (London: Pitman, 1982).

  3. L.E. Dickson, The collected mathematical papers. Volumes 1-5 (Chelsea Publishing Co.: New York, 1975).

  4. N. Dunford, J.C. Schwartz, Linear operators. (J. Wiley and Sons, Inc.: New York, 1966).

  5. Emch G.: Méchanique quantique quaternionienne et Relativité restreinte. Helv. Phys. Acta 36, 739–788 (1963)

    MathSciNet  MATH  Google Scholar 

  6. R. Engelking, General topology. (Heldermann: Berlin, 1989).

  7. J.E. Gilbert, M.A.M. Murray, Clifford algebras and Dirac operators in harmonic analysis. Cambr. studies in advanced Mathem. 26 (Cambr. Univ. Press: Cambridge, 1991).

  8. P.R. Girard, Quaternions, Clifford algebras and relativistic Physics. (Birkhäuser: Basel, 2007).

  9. K. Gürlebeck, W. Sprössig,Quaternionic analysis and elliptic boundary value problem. (Birkhäuser: Basel, 1990).

  10. F. Gürsey, C.-H. Tze, On the role of division, Jordan and related algebras in particle physics. (World Scientific Publ. Co.: Singapore, 1996).

  11. M. Junge, Q. Xu, Representation of certain homogeneous Hilbertian operator spaces and applications. Invent. Mathematicae 179: 1 (2010), 75-118.

    Google Scholar 

  12. R.V. Kadison, J.R. Ringrose, Fundamentals of the theory of operator algebras. (Acad. Press: New York, 1983).

  13. I.L. Kantor, A.S. Solodovnikov, Hypercomplex numbers. (Springer-Verlag: Berlin, 1989).

  14. R. Killip, B. Simon, Sum rules and spectral measures of Schrödinger operators with L 2 potentials. Annals of Mathematics 170: 2 (2009), 739-782.

    Google Scholar 

  15. R.S. Krausshar, J. Ryan, Some conformally flat spin manifolds, Dirac operators and automorphic forms. J. Math. Anal. Appl. 325 (2007), 359-376.

    Google Scholar 

  16. V.V. Kravchenko, On a new approach for solving Dirac equations with some potentials and Maxwell’s sytem in inhomogeoneous media. Operator Theory 121 (2001), 278-306.

    Google Scholar 

  17. K. Kuratowski, Topology. (Mir: Moscow, 1966).

  18. S.V. Ludkovsky, F. van Oystaeyen, Differentiable functions of quaternion variables. Bull. Sci. Math. (Paris). Ser. 2. 127 (2003), 755-796.

  19. S.V. Ludkovsky, Differentiable functions of Cayley-Dickson numbers and line integration. J. of Mathem. Sciences 141: 3 (2007), 1231-1298.

    Google Scholar 

  20. S.V. Ludkovsky. Algebras of operators in Banach spaces over the quaternion skew field and the octonion algebra. J. Mathem. Sciences 144: 4 (2008), 4301-4366.

    Google Scholar 

  21. S.V. Ludkovsky, Residues of functions of octonion variables. Far East Journal of Mathematical Sciences (FJMS), 39: 1 (2010), 65-104.

    Google Scholar 

  22. S.V. Ludkovsky, Analysis over Cayley-Dickson numbers and its applications. (LAP Lambert Academic Publishing: Saarbrn̎ucken, 2010).

  23. S.V. Ludkovsky, W. Sproessig. Ordered representations of normal and superdifferential operators in quaternion and octonion Hilbert spaces. Adv. Appl. Clifford Alg. 20: 2 (2010), 321-342.

  24. S.V. Ludkovsky, W. Sprössig, Spectral theory of super-differential operators of quaternion and octonion variables, Adv. Appl. Clifford Alg. 21: 1(2011), 165-191.

    Google Scholar 

  25. S.V. Ludkovsky, W. Sprössig, Spectral representations of operators in Hilbert spaces over quaternions and octonions, Complex Variables and Elliptic Equations 57: 12 (2012), 1301-1324, doi:10.1080/17476933.2010.538845.

  26. S.V. Ludkovsky. Integration of vector hydrodynamical partial differential equations over octonions. Complex Variables and Elliptic Equations, online, doi:10.1080/17476933.2011.598930, 31 pages (2011).

  27. S.V. Ludkovsky, Line integration of Dirac operators over octonions and Cayley-Dickson algebras. Computational Methods and Function Theory, 12:1 (2012), 279-306.

    Google Scholar 

  28. S.V. Ludkovsky, Operator algebras over Cayley-Dickson numbers. (LAP LAMBERT Academic Publishing AG & Co. KG: Saarbr¨ucken, 2011).

  29. F. van Oystaeyen, Algebraic geometry for associative algebras. Series ”Lect. Notes in Pure and Appl. Mathem.“ 232 (Marcel Dekker: New York, 2000).

  30. R.D. Schafer, An introduction to non-associative algebras. (Academic Press: New York, 1966).

  31. S. Zelditch, Inverse spectral problem for analytic domains, II: Z2-symmetric domains. Advances in Mathematics 170: 1 (2009), 205-269.

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Moscow State Technical Univ, MIREA av. Vernadsky, Moscow, 119454, Russian Federation

    Sergey V. Ludkowski

Authors
  1. Sergey V. Ludkowski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sergey V. Ludkowski.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ludkowski, S.V. Unbounded Normal Operators in Octonion Hilbert Spaces and Their Spectra. Adv. Appl. Clifford Algebras 23, 701–739 (2013). https://doi.org/10.1007/s00006-013-0393-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00006-013-0393-5

Keywords

Search

Navigation