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

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(A^m+n)=D(A^m)Aⁿ+A^mD(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(A^m)=A^mB-BA^m 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.

DOI: 10.3336/gm.52.2.04

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