Research Article
Characterizations of additive local Jordan derivations by action at idempotent operators
DOI:
10.2989/16073606.2026.2677476
Author(s):
Ting ZhangSchool of Mathematics and Statistics, Shanxi University, Taiyuan 030006, Shanxi, China, and Key Laboratory of Complex Systems and Data Science of Ministry of Education, Shanxi University, Taiyuan, China, Wei LuoSchool of Mathematics and Statistics, Shanxi University, Taiyuan 030006, Shanxi, China, and Key Laboratory of Complex Systems and Data Science of Ministry of Education, Shanxi University, Taiyuan, China, Xiaofei QiSchool of Mathematics and Statistics, Shanxi University, Taiyuan 030006, Shanxi, China, and Key Laboratory of Complex Systems and Data Science of Ministry of Education, Shanxi University, Taiyuan, China,
Abstract
Let H be an infinite-dimensional Hilbert space over the real or complex number field the algebra of all bounded linear operators on H . Assume that is an additive map. In this paper, we prove that δ is a local Jordan derivation if and only if δ(P ) = δ(P )P + P δ(P ) holds for all idempotent operators , if and only if δ is an inner derivation, that is, there exsits some such that δ(A) = AR − RA for all . As applications, we show that δ satisfies δ(A)B + Aδ(B) + δ(B)A + Bδ(A) = D for all with AB + BA = C , where are any two fixed operators if and only if there exsits some and some such that D = δ(C ) + µC and δ(A) = AR − RA + µA for all ; and give complete characterizations of δ satisfying δ(A)A + Aδ(A) = δ(P ) for all with A2
= P , where P = 0, I or a nontrivial idempotent operator with infinite-dimensional range and infinite- dimensional kernel. These generalize several known results.
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