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

Jianlian Cui, and Jinchuan Hou. "Linear maps preserving elements annihilated by the polynomial $XY-YX^{†}$." Studia Mathematica 174.2 (2006): 183-199. <http://eudml.org/doc/284911>.

@article{JianlianCui2006,
abstract = {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).},
author = {Jianlian Cui, Jinchuan Hou},
journal = {Studia Mathematica},
keywords = {indefinite inner product spaces; linear maps; zeros of polynomials; isomorphism},
language = {eng},
number = {2},
pages = {183-199},
title = {Linear maps preserving elements annihilated by the polynomial $XY-YX^\{†\}$},
url = {http://eudml.org/doc/284911},
volume = {174},
year = {2006},
}

TY - JOUR
AU - Jianlian Cui
AU - Jinchuan Hou
TI - Linear maps preserving elements annihilated by the polynomial $XY-YX^{†}$
JO - Studia Mathematica
PY - 2006
VL - 174
IS - 2
SP - 183
EP - 199
AB - 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).
LA - eng
KW - indefinite inner product spaces; linear maps; zeros of polynomials; isomorphism
UR - http://eudml.org/doc/284911
ER -