On certain equations in semiprime rings and standard operator algebras

Abstract

Abstract The purpose of this paper is to prove the following result which is related to a classical result of Chernoff. Let X be a real or complex Banach space, let {\mathcal{L}(X)} be the algebra of all bounded linear operators of X into itself and let {\mathcal{A}(X)\subset\mathcal{L}(X)} be a standard operator algebra. Suppose there exist linear mappings {\mathcal{H},\mathcal{G}\colon\mathcal{A(X)}\to\mathcal{L(X)}} satisfying the relations \displaystyle\mathcal{H}(\mathcal{A}^{m+n})=\mathcal{H}(\mathcal{A}^{m})% \mathcal{A}^{n}+\mathcal{A}^{m}\mathcal{G}(\mathcal{A}^{n}), \displaystyle\mathcal{G}(\mathcal{A}^{m+n})=\mathcal{G}(\mathcal{A}^{m})% \mathcal{A}^{n}+\mathcal{A}^{m}\mathcal{H}(\mathcal{A}^{n}) for all {\mathcal{A}\in\mathcal{A(X)}} and some fixed integers {m,n\geq 1} . Then there exists {\mathcal{B}\in\mathcal{L(X)}} , such that {\mathcal{H(A)}=\mathcal{AB}-\mathcal{BA}} for all {\mathcal{A}\in\mathcal{F(X)}} , where {\mathcal{F(X)}} denotes the ideal of all finite rank operators in {\mathcal{L}(X)} , and {\mathcal{H}(\mathcal{A}^{m})=\mathcal{A}^{m}\mathcal{B}-\mathcal{B}\mathcal{A% }^{m}} for all {\mathcal{A}\in\mathcal{A(X)}} .