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Linear maps preserving elements annihilated by the polynomial X Y - Y X

Jianlian CuiJinchuan 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 CuiJinchuan 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...

Linear maps Lie derivable at zero on 𝒥-subspace lattice algebras

Xiaofei QiJinchuan Hou — 2010

Studia Mathematica

A linear map L on an algebra is said to be Lie derivable at zero if L([A,B]) = [L(A),B] + [A,L(B)] whenever [A,B] = 0. It is shown that, for a 𝒥-subspace lattice ℒ on a Banach space X satisfying dim K ≠ 2 whenever K ∈ 𝒥(ℒ), every linear map on ℱ(ℒ) (the subalgebra of all finite rank operators in the JSL algebra Alg ℒ) Lie derivable at zero is of the standard form A ↦ δ (A) + ϕ(A), where δ is a generalized derivation and ϕ is a center-valued linear map. A characterization of linear maps Lie derivable...

Jordan isomorphisms and maps preserving spectra of certain operator products

Jinchuan HouChi-Kwong LiNgai-Ching Wong — 2008

Studia Mathematica

Let ₁, ₂ be (not necessarily unital or closed) standard operator algebras on locally convex spaces X₁, X₂, respectively. For k ≥ 2, consider different products T T k on elements in i , which covers the usual product T T k = T T k and the Jordan triple product T₁ ∗ T₂ = T₂T₁T₂. Let Φ: ₁ → ₂ be a (not necessarily linear) map satisfying σ ( Φ ( A ) Φ ( A k ) ) = σ ( A A k ) whenever any one of A i ’s has rank at most one. It is shown that if the range of Φ contains all rank one and rank two operators then Φ must be a Jordan isomorphism multiplied by a root...

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