Preservation of the Norms of Linear Operators Acting on Some Quaternionic Function Spaces

  • M. Elena Luna-Elizarrarás
  • Michael Shapiro
Part of the Operator Theory: Advances and Applications book series (OT, volume 157)


There are considered real and quaternionic versions of some classical linear spaces, such as Lebesgue spaces, the spaces of continuous functions, etc., as well as linear operators acting on them. We prove that, given a real linear bounded operator on one of those spaces its quaternionic extension keeps being bounded and its norm does not change.


Quaternionic function spaces quaternionic linear operators norms of quaternionic extensions of linear operators 


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  1. [Ad]
    Adler. Quaternionic Quantum Mechanics and Quantum Fields. Oxford University Press, 586 pp., 1995.Google Scholar
  2. [AgKu]
    S. Agrawal, S.H. Kulkarni, Dual spaces of quaternion normed linear spaces and reflexivity. J. Anal. 8 (2000), 79–90.Google Scholar
  3. [AS]
    D. Alpay, M.V. Shapiro, Reproducing Kernel Quaternionic Pontryagin Spaces. Integral equations and operator theory 50(4) (2004), 431–476.CrossRefMathSciNetGoogle Scholar
  4. [ASV]
    D. Alpay, M. Shapiro, D. Volok, Espaces de Branges Rovnyak et fonctions de Schur: le cas hyper-analytique. Comtes Rendus de l’Académie des Sciences–Mathématique, Ser. I, 2004, v. 338, 437–442.Google Scholar
  5. [BDS]
    F. Brackx, R. Delanghe, F. Sommen, Clifford analysis. London: Pitman Res. Notes in Math., v. 76, 308 pp.Google Scholar
  6. [De]
    A. Defant, Best constants for the norm of the Complexification of Operators Between Lp-spaces. Lecture Notes in Pure and Applied Mathematics 150 (1994), 173–180.Google Scholar
  7. [DSS]
    R. Delanghe, F. Sommen, V. Soucek, Clifford algebra and spinor-valued functions. Amsterdam: Kluwer Acad. Publ, 1992, 485 pp.Google Scholar
  8. [DuSch]
    N. Dunford, J.T. Schwartz, Linear Operators, Part I. Interscience Publishers, Inc., New York, 1957.Google Scholar
  9. [FIP]
    T. Figiel, T. Iwaniec, A. Pelczyński, Computing norms and critical exponents of some operators in Lp-spaces. Stud. Math. 79 (1984), No. 3, 227–274.Google Scholar
  10. [GaMa]
    J. Gash, L. Maligranda, On Vector-valued Inequalities of the Marcinkiewicz-Zygmund. Herz and Krivine Type. Math. Nachr 167 (1994), 95–129.Google Scholar
  11. [GüSp1]
    K. Gürlebeck, W. Sprößig, Quaternionic analysis and elliptic boundary value problems. Berlin: Akademie-Verlag, 1989.Google Scholar
  12. [GüSp2]
    K. Gürlebeck, W. Sprößig, Quaternionic and Clifford Calculus for Physicists and Engineers. Wiley & Sons, 1997.Google Scholar
  13. [Krl]
    J.I. Krivine, Sur la Complexification des Opérateurs de L dans L1. C.R. Acad. Sci. Paris 284 (1977), 377–379.Google Scholar
  14. [Kr2]
    J.I. Krivine, Constantes de Grothendieck et Fonctions de Type Positif sur les Sphères. Adv. Math. 31 (1979), 16–30.CrossRefGoogle Scholar
  15. [Ri]
    M. Riesz, Sur les Maxima des Formes Bilinéaires et sur les Fonctionelles Linéaires. Acta Math. 49 (1926), 465–497.Google Scholar
  16. [Sh]
    M.V. Shapiro, Structure of the quaternionic modules and some properites of the involutive operators. J. of Natural Geometry (London) 1, # 1 (1992), 9–37.Google Scholar
  17. [ShVal]
    M.V. Shapiro, N.L. Vasilevski, Quaternionic ψ-hyperholomorphic functions, singular integral operators and boundary value problems. I. ψ-hyperholomorphic function theory. Complex Variables. Theory and Applications 27 (1995), 17–46.Google Scholar
  18. [ShVa2]
    M.V. Shapiro, N.L. Vasilevski, Quaternionic ψ-hyperholomorphic functions. singular integral operators and boundary value problems. II. Algebras of singular integral operators and Riemann-type boundary value problems. Complex Variables. Theory and Applications 27 (1995), 67–96.Google Scholar
  19. [Shar]
    C.S. Sharma, Complex structure on a real Hilbert space and symplectic structure on a complex Hilbert space. J. Math. Phys 29, # 5 (1988), 1069–1078.CrossRefGoogle Scholar
  20. [SharAl1]
    C.S. Sharma, D.F. Almeida, Semilinear operators. J. Math. Phys. 29, # 11 (1988), 2411–2420.CrossRefGoogle Scholar
  21. [SharAl2]
    C.S. Sharma, D.F. Almeida, Additive functionals and operators on a quaternionic Hilbert space. J. Math. Phys. 30, # 2 (1989), 369–375.CrossRefGoogle Scholar
  22. [So]
    J. Sokolowski, On the Norm of the Complex Extension of the Linear Operator. Mat. Issled. 54 (1980), 152–154 (in Russian).Google Scholar
  23. [St]
    S.B. Stechkin, On the Best Lacunary System of Functions. Izv. Acad. Nauk SSSR, Ser. Mat. 25 (1961), 357–366 (in Russian).Google Scholar
  24. [Ve]
    I.E. Verbitski, Some Relations Between the Norm of an Operator and that of its Complex Extension. Mat. Issled. 42 (1976), 3–12 (in Russian).Google Scholar
  25. [VeSe]
    I.E. Verbitski, P.P. Sereda, About the norm of the complex extension of an operator. Mat. Issled. 37 (1975), 201–206 (in Russian).Google Scholar
  26. [Zy]
    A. Zygmund, Trigonometric Series. Volume I. Cambridge University Press, 1968.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2005

Authors and Affiliations

  • M. Elena Luna-Elizarrarás
    • 1
  • Michael Shapiro
    • 1
  1. 1.Departamento de MatemáticasE.S.F.M. del I.P.N.México, D.F.México

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