Characterization of Lie centralizable mappings on B(X)

TL;DR

This paper characterizes additive Lie centralizable mappings on the algebra of bounded linear operators on a Banach space, showing they have a specific form involving scalar multiplication and a central additive map under certain conditions.

Contribution

It provides a new characterization of Lie centralizable mappings on B(X), extending understanding of their structure when centered at operators with specific properties.

Findings

Mappings are of the form f(A)=kA+h(A) with h commuting with certain operators

Identifies conditions under which these mappings are additive and centralizable

Extends previous results on Lie derivations in operator algebras

Abstract

Assume that B(X) is the algebra of all bounded linear operators on a complex Banach space X, and let W in B(X) is such that cl(W(X)) is not equal to X or W=zI, where z is a complex number and I is the identity operator. We show that if f: B(X) --> B(X) is an additive mapping Lie centralizable at W, then f(A)=kA+h(A) for all A in B(X), where k is a complex number and h:B(X)--> CI is an additive mapping such that h([A,B])=0 for all A,B in B(X) with AB=W.