Wyniki wyszukiwania

1

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

100%

Qi X., Hou J.

Studia Mathematica

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2010

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tom 197

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nr 2

157-169

EN

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 at zero on Alg ℒ is also obtained, which are not of the above standard form in general.

2

Linear maps preserving elements annihilated by the polynomial $XY-YX^{†}$

100%

Cui J., Hou J.

Studia Mathematica

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2006

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tom 174

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nr 2

183-199

EN

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 $AB = BA^{†}$ if and only if there exist a nonzero real number c and a generalized indefinite unitary operator U ∈ ℬ(H,K) such that $Φ(A) = cUAU^{†}$ for all A ∈ ℬ(H).

3

The spectrally bounded linear maps on operator algebras

100%

Cui J., Hou J.

Studia Mathematica

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2002

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tom 150

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nr 3

261-271

EN

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 complex Hilbert spaces, is either an isomorphism or an anti-isomorphism multiplied by a nonzero complex number. If Φ is not injective, then Φ vanishes at all compact operators.

4

Jordan isomorphisms and maps preserving spectra of certain operator products

81%

Hou J., Li C.-K., Wong N.-C.

Studia Mathematica

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2008

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tom 184

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nr 1

31-47

EN

Let 𝓐₁, 𝓐₂ be (not necessarily unital or closed) standard operator algebras on locally convex spaces X₁, X₂, respectively. For k ≥ 2, consider different products $T₁ ∗ ⋯ ∗ T_{k}$ on elements in $𝓐_{i}$, which covers the usual product $T₁ ∗ ⋯ ∗ T_{k} = T₁ ⋯ T_{k}$ and the Jordan triple product T₁ ∗ T₂ = T₂T₁T₂. Let Φ: 𝓐₁ → 𝓐₂ be a (not necessarily linear) map satisfying $σ(Φ(A₁) ∗ ⋯ ∗ Φ(A_{k})) = σ(A₁ ∗ ⋯ ∗ A_{k})$ 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 of unity. Similar results for self-adjoint operators acting on Hilbert spaces are obtained.

5

Maps preserving numerical radius distance on C*-algebras

81%

Bai Z., Hou J., Xu Z.

Studia Mathematica

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2004

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tom 162

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nr 2

97-104

EN

We characterize surjective nonlinear maps Φ between unital C*-algebras 𝒜 and ℬ that satisfy w(Φ(A)-Φ(B))) = w(A-B) for all A,B ∈ 𝒜 under a mild condition that Φ(I) - Φ(0) belongs to the center of ℬ, where w(A) is the numerical radius of A and I is the unit of 𝒜.

Ograniczanie wyników

5 Studia Mathematica

5 Hou J.

2 Cui J.

1 Bai Z.

1 Li C.-K.

1 Qi X.

1 Wong N.-C.

1 Xu Z.

1 2010

1 2008

1 2006

1 2004

1 2002

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

Linear maps preserving elements annihilated by the polynomial $XY-YX^{†}$

The spectrally bounded linear maps on operator algebras

Jordan isomorphisms and maps preserving spectra of certain operator products

Maps preserving numerical radius distance on C*-algebras