Jordan Product and ∂-Spectrum Preservers
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Consider an infinite-dimentional Banach space denoted as X, and let L(X) denotes the algebra of all bounded linear operators on X. Additionally, σ(A) denotes the spectrum of A ∈ L(X) and ∂(σ(A)) stands for the boundary of σ(A). A map Δ : L(X) → 2C is called a ∂-spectrum if ∂(σ(A)) ⊆ Δ(A) ⊆ σ(A) for all A ∈ L(X). In this paper, we characterize all surjective maps φ on L(X) preseving the ∂-spectrum of Jordan product of operators, i.e., Δ(φ(A) ◦φ(B)) = Δ(A◦B), for all A,B ∈ L(X), where A◦B = AB+BA.
Abstract
Consider an infinite-dimentional Banach space denoted as X, and let L(X) denotes the algebra of all bounded linear operators on X. Additionally, σ(A) denotes the spectrum of A ∈ L(X) and ∂(σ(A)) stands for the boundary of σ(A). A map Δ : L(X) → 2C is called a ∂-spectrum if ∂(σ(A)) ⊆ Δ(A) ⊆ σ(A) for all A ∈ L(X). In this paper, we characterize all surjective maps φ on L(X) preseving the ∂-spectrum of Jordan product of operators, i.e., Δ(φ(A) ◦φ(B)) = Δ(A◦B), for all A,B ∈ L(X), where A◦B = AB+BA.