Glasnik Matematicki, Vol. 52, No. 2 (2017), 241-246.

ON CERTAIN EQUATION RELATED TO DERIVATIONS ON STANDARD OPERATOR ALGEBRAS AND SEMIPRIME RINGS

Irena Kosi-Ulbl

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

Abstract.   In this paper we prove the following result, which is related to a classical result of Chernoff. Let X be a real or complex Banach space, let A(X) be a standard operator algebra on X and let L (X) be an algebra of all bounded linear operators on X. Suppose we have a linear mapping D: A(X) → L (X) satisfying the relation D(Am+n)=D(Am)An+AmD(An) for all A A(X) and some fixed integers m≥1,n≥1. In this case there exists B L (X), such that D(A)=AB-BA holds for all A F(X), where F (X) denotes the ideal of all finite rank operators in L (X). Besides, D(Am)=AmB-BAm is fulfilled for all A A(X).

2010 Mathematics Subject Classification.   16N60, 39B05, 46K15.

Key words and phrases.   Prime ring, semiprime ring, Banach space, standard operator algebra, derivation, Jordan derivation.


Full text (PDF) (free access)

DOI: 10.3336/gm.52.2.04


References:

  1. M. Brešar and J. Vukman, Jordan derivations on prime rings, Bull. Austral. Math. Soc. 37 (1988), 321-322.
    MathSciNet     CrossRef

  2. M. Brešar, Jordan derivations on semiprime rings, Proc. Amer. Math. Soc. 104 (1988), 1003-1006.
    MathSciNet     CrossRef

  3. P. R. Chernoff, Representations, automorphisms, and derivations of some operator algebras, J. Functional Analysis 12 (1973), 275-289.
    MathSciNet     CrossRef

  4. J. M. Cusack, Jordan derivations on rings, Proc. Amer. Math. Soc. 53 (1975), 321-324.
    MathSciNet     CrossRef

  5. I. N. Herstein, Jordan derivations of prime rings, Proc. Amer. Math. Soc. 8 (1957), 1104-1110.
    MathSciNet     CrossRef

  6. N. Širovnik and J. Vukman, On functional equation related to derivations and bicircular projections, Oper. Matrices 8 (2014), 849-860.
    MathSciNet     CrossRef

  7. J. Vukman, On automorphisms and derivations of operator algebras, Glas. Mat. Ser. III 19(39) (1984), 135-138.
    MathSciNet    

Glasnik Matematicki Home Page