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

100%

Cui J., Hou J.

Studia Mathematica

2006

tom 174

nr 2

183-199

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).

The spectrally bounded linear maps on operator algebras

100%

Cui J., Hou J.

Studia Mathematica

2002

tom 150

nr 3

261-271

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.

Ograniczanie wyników

2 Studia Mathematica

2 Cui J.

2 Hou J.

1 2006

1 2002

Wyniki wyszukiwania

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

The spectrally bounded linear maps on operator algebras