Glasnik Matematicki, Vol. 50, No. 2 (2015), 363-371.

ON CERTAIN IDENTITY RELATED TO JORDAN *-DERIVATIONS

Irena Kosi-Ulbl and Joso Vukman

Faculty of Mechanical Engineering, University of Maribor, Smetanova 17, 2000 Maribor, Slovenia
e-mail: irena.kosi@um.si

Koroška cesta 57, 2000 Maribor, Slovenia
e-mail: joso.vukman@guest.um.si

Abstract.   In this paper we prove the following result. Let H be a real or complex Hilbert space, let L(H) be the algebra of all bounded linear operators on H and let A(H) ⊆ L(H) be a standard operator algebra. Suppose we have an additive mapping D:A(H) → L(H) satisfying the relation D(An)=D(A)A* n-1+AD(An-2)A* +An-1D(A) for all A A(H) and some fixed integer n>1. In this case there exists a unique B L(H) such that D(A)=BA*-AB holds for all A A(H).

2010 Mathematics Subject Classification.   16W10, 46K15, 39B05.

Key words and phrases.   Ring, ring with involution, prime ring, semiprime ring, Hilbert space, standard operator algebra, *-derivation, Jordan *-derivation.

Full text (PDF) (free access)

DOI: 10.3336/gm.50.2.07

References:

  1. M. Brešar and J. Vukman, On some additive mappings in rings with involution, Aequationes Math. 38 (1989), 178-185.
    MathSciNet     CrossRef

  2. M. Brešar and B. Zalar, On the structure of Jordan *-derivations, Colloq. Math. 63 (1997), 163-171.
    MathSciNet    

  3. H. G. Dales, Automatic continuity: a survey, Bull. Lond. Math. Soc. 10 (1978), 129-183.
    MathSciNet     CrossRef

  4. M. G. Dales, Banach algebras and automatic continuity, Calderon Press, Oxford, 2001.
    MathSciNet    

  5. D. Ilišević, Equations arising from Jordan *-derivation pairs, Aequationes Math. 67 (2004), 236-240.
    MathSciNet     CrossRef

  6. D. Ilišević, On quadratic functionals from *-rings to a class of Banach *-algebras, in Proceedings of the Postgraduate School and Conference Functional Analysis VIII, Aarhus Univ., Aarhus, 2004, 110-115.
    MathSciNet    

  7. D. Ilišević, Quadratic functionals on modules over *-rings, Studia Sci. Math. Hungar. 42 (2005), 95-105.
    MathSciNet     CrossRef

  8. D. Ilišević, Quadratic functionals on modules over alternative *-rings, Studia Sci. Math. Hungar. 42 (2005), 459-469.
    MathSciNet     CrossRef

  9. S. Kurepa, The Cauchy functional equation and scalar product in vector spaces, Glasnik Mat. Fiz.-Astronom. Ser. II Društvo Mat. Fiz. Hrvatske 19 (1964), 23-26.
    MathSciNet    

  10. S. Kurepa, Quadratic and sesquilinear functionals, Glasnik Mat. Fiz.-Astronom. Ser. II Društvo Mat. Fiz. Hrvatske 20 (1965), 79-92.
    MathSciNet    

  11. L. Molnár, Jordan *-derivation pairs on a complex *-algebras, Aequationes Math. 54 (1997), 44-55.
    MathSciNet     CrossRef

  12. A. M. Sinclair, Automatic continuity of linear operators, Cambridge University Press, Cambridge, London, New York and Melbourne, 1976.
    MathSciNet     CrossRef

  13. P. Šemrl, On quadratic functionals, Bull. Aust. Math. Soc. 37 (1988), 27-28.
    MathSciNet     CrossRef

  14. P. Šemrl, Quadratic and quasi-quadratic functionals, Proc. Amer. Math. Soc. 119 (1993), 1105-1113.
    MathSciNet     CrossRef

  15. P. Šemrl, Jordan *-derivations of standard operator algebras, Proc. Amer. Math. Soc. 120 (1994), 515-518.
    MathSciNet     CrossRef

  16. P. Vrbová, Quadratic functionals and bilinear forms, Časopis Pest. Mat. 98 (1973), 159-161.
    MathSciNet    

  17. J. Vukman, A result concerning additive functions in hermitian Banach *-algebras and an application, Proc. Amer. Math. Soc. 91 (1984), 367-372.
    MathSciNet     CrossRef

  18. J. Vukman, Some results concerning the Cauchy functional equation in certain Banach algebras, Bull. Aust. Math. Soc. 31 (1985), 137-144.
    MathSciNet     CrossRef

  19. J. Vukman, Some functional equations in Banach algebras and an application, Proc. Amer. Math. Soc. 100 (1987), 133-136.
    MathSciNet     CrossRef

  20. J. Vukman, A note on Jordan *-derivations in semiprime rings with involution, Int. Math. Forum 1 (2006), 617-622.
    MathSciNet    

  21. D. Yang, Jordan *-derivation pairs on standard operator algebras and related results, Colloq. Math. 102 (2005), 137-145.
    MathSciNet     CrossRef

  22. B. Zalar, Jordan-von Neumann theorem for Saworotnow's generalized Hilbert space, Acta Math. Hungar. 69 (1995), 301-325.
    MathSciNet     CrossRef

  23. B. Zalar, Jordan *-derivation pairs and quadratic functionals on modules over *-rings, Aequationes Math. 54 (1997), 31-43.
    MathSciNet     CrossRef
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