#### 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(A*^{m+n})=D(A^{m})A^{n}+A^{m}D(A^{n}) 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^{m}B-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.

**Full text (PDF)** (access from subscribing institutions only)
DOI: 10.3336/gm.52.2.04

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