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On certain equations in semiprime rings and standard operator algebras

Authors
  • Rehman, Nadeem ur
Type
Published Article
Journal
Advances in Pure and Applied Mathematics
Publisher
De Gruyter
Publication Date
Aug 08, 2018
Volume
10
Issue
3
Pages
241–250
Identifiers
DOI: 10.1515/apam-2018-0033
Source
De Gruyter
Keywords
License
Yellow

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 β„’ ⁒ ( X ) {\mathcal{L}(X)} be the algebra of all bounded linear operators of X into itself and let π’œ ⁒ ( X ) βŠ‚ β„’ ⁒ ( X ) {\mathcal{A}(X)\subset\mathcal{L}(X)} be a standard operator algebra. Suppose there exist linear mappings β„‹ , 𝒒 : π’œ ⁒ ( 𝒳 ) β†’ β„’ ⁒ ( 𝒳 ) {\mathcal{H},\mathcal{G}\colon\mathcal{A(X)}\to\mathcal{L(X)}} satisfying the relations β„‹ ⁒ ( π’œ m + n ) = β„‹ ⁒ ( π’œ m ) ⁒ π’œ n + π’œ m ⁒ 𝒒 ⁒ ( π’œ n ) , \displaystyle\mathcal{H}(\mathcal{A}^{m+n})=\mathcal{H}(\mathcal{A}^{m})% \mathcal{A}^{n}+\mathcal{A}^{m}\mathcal{G}(\mathcal{A}^{n}), 𝒒 ⁒ ( π’œ m + n ) = 𝒒 ⁒ ( π’œ m ) ⁒ π’œ n + π’œ m ⁒ β„‹ ⁒ ( π’œ 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 π’œ ∈ π’œ ⁒ ( 𝒳 ) {\mathcal{A}\in\mathcal{A(X)}} and some fixed integers m , n β‰₯ 1 {m,n\geq 1} . Then there exists ℬ ∈ β„’ ⁒ ( 𝒳 ) {\mathcal{B}\in\mathcal{L(X)}} , such that β„‹ ⁒ ( π’œ ) = π’œ ⁒ ℬ - ℬ ⁒ π’œ {\mathcal{H(A)}=\mathcal{AB}-\mathcal{BA}} for all π’œ ∈ β„± ⁒ ( 𝒳 ) {\mathcal{A}\in\mathcal{F(X)}} , where β„± ⁒ ( 𝒳 ) {\mathcal{F(X)}} denotes the ideal of all finite rank operators in β„’ ⁒ ( X ) {\mathcal{L}(X)} , and β„‹ ⁒ ( π’œ m ) = π’œ m ⁒ ℬ - ℬ ⁒ π’œ m {\mathcal{H}(\mathcal{A}^{m})=\mathcal{A}^{m}\mathcal{B}-\mathcal{B}\mathcal{A% }^{m}} for all π’œ ∈ π’œ ⁒ ( 𝒳 ) {\mathcal{A}\in\mathcal{A(X)}} .

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