Generalized derivable mappings and centralizable mappings on B(X)

Abstract

Let X be a Banach space and $B(X)$ be the algebra of all bounded linear operators on X. Firstly, we prove that every derivable mapping at a nonzero element in $B(X)$ is a derivation. As a corollary, we show that every generalized derivable mapping on $B(X)$ is a general derivation. Secondly, we prove that every centralizable mapping on $B(X)$ is a centralizer.

KEYWORDS:

• Derivation
• derivable mapping
• centralizer
• centralizable mapping

MR(2010) SUBJECT CLASSIFICATIONS:

• 47B47
• 47L10

Acknowledgments

We thank the referees for their time and comments. We thank Maxine Garcia, PhD, from Liwen Bianji (Edanz) (www.liwenbianji.cn/) for editing the English text of a draft of this manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This paper was partially supported by the National Natural Science Foundation of China (Grant Nos. 11801005 and 11801342). The first author was also partially supported by the Startup Foundation of Anhui Polytechnic University (Grant No. 2017YQQ017) and the third author was also partially supported by the Natural Science Foundation of Shaanxi Province (Grant No. 2020JQ-693).