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

Jianlian Cui; Jinchuan Hou

Linear maps preserving elements annihilated by the polynomial

X Y - Y X^{†}

Jianlian Cui ; Jinchuan Hou

Studia Mathematica, Tome 173 (2006), p. 183-199 / Harvested from The Polish Digital Mathematics Library

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Résumé

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).

Publié le : 2006-01-01

Zbl 1091.47030

EUDML-ID : urn:eudml:doc:284911

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     title = {Linear maps preserving elements annihilated by the polynomial $XY-YX^{\dag }$
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Jianlian Cui; Jinchuan Hou. Linear maps preserving elements annihilated by the polynomial $XY-YX^{†}$
            . Studia Mathematica, Tome 173 (2006) pp. 183-199. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm174-2-5/