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On certain equations in semiprime rings and standard operator algebras

  • Nadeem ur Rehman ORCID logo EMAIL logo
Published/Copyright: August 8, 2018

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 ℒ⁒(X) be the algebra of all bounded linear operators of X into itself and let π’œβ’(X)βŠ‚β„’β’(X) be a standard operator algebra. Suppose there exist linear mappings β„‹,𝒒:π’œβ’(𝒳)→ℒ⁒(𝒳) satisfying the relations

ℋ⁒(π’œm+n)=ℋ⁒(π’œm)β’π’œn+π’œm⁒𝒒⁒(π’œn),
𝒒⁒(π’œm+n)=𝒒⁒(π’œm)β’π’œn+π’œm⁒ℋ⁒(π’œn)

for all π’œβˆˆπ’œβ’(𝒳) and some fixed integers m,nβ‰₯1. Then there exists β„¬βˆˆβ„’β’(𝒳), such that ℋ⁒(π’œ)=π’œβ’β„¬-β„¬β’π’œ for all π’œβˆˆβ„±β’(𝒳), where ℱ⁒(𝒳) denotes the ideal of all finite rank operators in ℒ⁒(X), and ℋ⁒(π’œm)=π’œm⁒ℬ-β„¬β’π’œm for all π’œβˆˆπ’œβ’(𝒳).

MSC 2010: 16W25; 16W20; 16N60

Acknowledgements

The author is greatly indebted to the referee for his/her valuable suggestions which have improved the paper immensely.

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Received: 2018-02-21
Revised: 2018-04-19
Accepted: 2018-07-11
Published Online: 2018-08-08
Published in Print: 2019-07-01

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