Maps Preserving Peripheral Spectrum of Jordan Products of Operators
نویسنده
چکیده
Let A and B be (not necessarily unital or closed) standard operator algebras on complex Banach spaces X and Y , respectively. For a bounded linear operator A on X, the peripheral spectrum σπ(A) of A is defined by σπ(A) = {z ∈ σ(A) : |z| = maxw∈σ(A) |w|}, where σ(A) denotes the spectrum of A. Assume that Φ : A → B is a map and the range of Φ contains all operators with rank at most two. It is proved that the map Φ satisfies the condition that σπ(Φ(A)Φ(B) + Φ(B)Φ(A)) = σπ(AB + BA) for all A,B ∈ A if and only if either there exists an invertible operator T ∈ B(X,Y ) such that Φ(A) = εTAT−1 for every A ∈ A; or X and Y are reflexive and there exists an invertible operator T ∈ B(X∗, Y ) such that Φ(A) = εTA∗T−1 for every A ∈ A, where ε ∈ {1,−1}. Furthermore, the same conclusion holds if A and B are replaced by standard real Jordan algebras of self-adjoint operators on complex Hilbert spaces. If X and Y are complex Hilbert space, we characterize also maps preserving the peripheral spectrum of the product AB∗ + B∗A, and prove that such maps are of the form A 7→ γUAU∗ or A 7→ γUAtU∗, where U ∈ B(X,Y ) is a unitary operator and γ ∈ C with |γ| = 1, A denotes the transpose of A for an arbitrary but fixed orthonormal basis of X.
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تاریخ انتشار 2011