Characterization of (α,α)-derivation on B(X)

2024/03/05 by Quanyuan Chen, Chen, Quanyuan, Y G Li +1
Physics and Astronomy · Mathematics · #Advanced Differential Geometry Research #Advanced Topics in Algebra #Advanced Differential Equations and Dynamical Systems

paper · pdf · doi:10.48550/arxiv.2403.02815

Abstract

Let X be a Banach algebra and B(X) be the set of all bounded linear operators on X. Suppose that α: B(X) → B(X) is an automorphism. We say that a mapping δ from B(X) into itself is derivable at G ∈ B(X) if δ(G) = α(A)δ(B) + δ(A)α(B) for all A, B ∈ B(X) with AB = G. We say that an element G ∈ B(X) is an (α,α)-all derivable point of B(X) if every (α,α)-derivable mapping δ at G is an (α,α)-derivation. In this paper, we show that every (α,α)-derivable mapping at a nonzero element in B(X) is an (α,α)-derivation.

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