Linear maps preserving A -unitary operators

Abdellatif Chahbi; Samir Kabbaj; Ahmed Charifi

Mathematica Bohemica (2016)

  • Volume: 141, Issue: 1, page 59-70
  • ISSN: 0862-7959

Abstract

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Let ℋ be a complex Hilbert space, A a positive operator with closed range in ℬ ( ℋ ) and ℬ A ( ℋ ) the sub-algebra of ℬ ( ℋ ) of all A -self-adjoint operators. Assume φ : ℬ A ( ℋ ) onto itself is a linear continuous map. This paper shows that if φ preserves A -unitary operators such that φ ( I ) = P then ψ defined by ψ ( T ) = P φ ( P T ) is a homomorphism or an anti-homomorphism and ψ ( T ♯ ) = ψ ( T ) ♯ for all T ∈ ℬ A ( ℋ ) , where P = A + A and A + is the Moore-Penrose inverse of A . A similar result is also true if φ preserves A -quasi-unitary operators in both directions such that there exists an operator T satisfying P φ ( T ) = P .

How to cite

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Chahbi, Abdellatif, Kabbaj, Samir, and Charifi, Ahmed. "Linear maps preserving $A$-unitary operators." Mathematica Bohemica 141.1 (2016): 59-70. <http://eudml.org/doc/276755>.

@article{Chahbi2016,
abstract = {Let $\mathcal \{H\}$ be a complex Hilbert space, $A$ a positive operator with closed range in $\mathcal \{B\}(\mathcal \{H\})$ and $\mathcal \{B\}_\{A\}(\mathcal \{H\})$ the sub-algebra of $\mathcal \{B\}(\mathcal \{H\})$ of all $A$-self-adjoint operators. Assume $\phi \colon \mathcal \{B\}_\{A\}(\mathcal \{H\})$ onto itself is a linear continuous map. This paper shows that if $\phi $ preserves $A$-unitary operators such that $\phi (I)=P$ then $\psi $ defined by $\psi (T)=P\phi (PT)$ is a homomorphism or an anti-homomorphism and $\psi (T^\{\sharp \})=\psi (T)^\{\sharp \}$ for all $T \in \mathcal \{B\}_\{A\}(\mathcal \{H\})$, where $P=A^\{+\}A$ and $A^\{+\}$ is the Moore-Penrose inverse of $A$. A similar result is also true if $\phi $ preserves $A$-quasi-unitary operators in both directions such that there exists an operator $T$ satisfying $P\phi (T)=P$.},
author = {Chahbi, Abdellatif, Kabbaj, Samir, Charifi, Ahmed},
journal = {Mathematica Bohemica},
keywords = {linear preserver problem; semi-inner product},
language = {eng},
number = {1},
pages = {59-70},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Linear maps preserving $A$-unitary operators},
url = {http://eudml.org/doc/276755},
volume = {141},
year = {2016},
}

TY - JOUR
AU - Chahbi, Abdellatif
AU - Kabbaj, Samir
AU - Charifi, Ahmed
TI - Linear maps preserving $A$-unitary operators
JO - Mathematica Bohemica
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 141
IS - 1
SP - 59
EP - 70
AB - Let $\mathcal {H}$ be a complex Hilbert space, $A$ a positive operator with closed range in $\mathcal {B}(\mathcal {H})$ and $\mathcal {B}_{A}(\mathcal {H})$ the sub-algebra of $\mathcal {B}(\mathcal {H})$ of all $A$-self-adjoint operators. Assume $\phi \colon \mathcal {B}_{A}(\mathcal {H})$ onto itself is a linear continuous map. This paper shows that if $\phi $ preserves $A$-unitary operators such that $\phi (I)=P$ then $\psi $ defined by $\psi (T)=P\phi (PT)$ is a homomorphism or an anti-homomorphism and $\psi (T^{\sharp })=\psi (T)^{\sharp }$ for all $T \in \mathcal {B}_{A}(\mathcal {H})$, where $P=A^{+}A$ and $A^{+}$ is the Moore-Penrose inverse of $A$. A similar result is also true if $\phi $ preserves $A$-quasi-unitary operators in both directions such that there exists an operator $T$ satisfying $P\phi (T)=P$.
LA - eng
KW - linear preserver problem; semi-inner product
UR - http://eudml.org/doc/276755
ER -

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