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Linear maps preserving elements annihilated by the polynomial $X Y - Y X^{†}$

Jianlian Cui; Jinchuan Hou — 2006

Studia Mathematica

Let H and K be complex complete indefinite inner product spaces, and ℬ(H,K) (ℬ(H) if K = H) the set of all bounded linear operators from H into K. For every T ∈ ℬ(H,K), denote by $T^{†}$ the indefinite conjugate of T. Suppose that Φ: ℬ(H) → ℬ(K) is a bijective linear map. We prove that Φ satisfies $Φ (A) Φ (B) = Φ (B) Φ {(A)}^{†}$ for all A, B ∈ ℬ(H) with $A B = B A^{†}$ if and only if there exist a nonzero real number c and a generalized indefinite unitary operator U ∈ ℬ(H,K) such that $Φ (A) = c U A U^{†}$ for all A ∈ ℬ(H).

The spectrally bounded linear maps on operator algebras

Jianlian Cui; Jinchuan Hou — 2002

Studia Mathematica

We show that every spectrally bounded linear map Φ from a Banach algebra onto a standard operator algebra acting on a complex Banach space is square-zero preserving. This result is used to show that if Φ₂ is spectrally bounded, then Φ is a homomorphism multiplied by a nonzero complex number. As another application to the Hilbert space case, a classification theorem is obtained which states that every spectrally bounded linear bijection Φ from ℬ(H) onto ℬ(K), where H and K are infinite-dimensional...