PRODUCT OF OPERATORS AND ∂-SPECTRUM PRESERVERS
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Abstract
Consider an infinite-dimentional Banach space denoted as X, and designate B(X) as the algebra of all bounded linear operators on X. Moreover, let σ(A) denote the spectrum of A ∈ B (X), and ∂(σ(A)) indicate the boundary of σ(A). A map ∆ : B(X) → 2C is termed a ∂-spectrum if ∂(σ(A)) ⊆ ∆(A) ⊆ σ(A) for all A ∈ B(X). In this paper, we characterize all surjective maps ϕ1 and ϕ2 on B(X) satisfying ∆(ϕ1(A)ϕ2(B)) = ∆(AB) for all A, B ∈ B(X).
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