| Title:
|
Linear maps preserving $A$-unitary operators (English) |
| Author:
|
Chahbi, Abdellatif |
| Author:
|
Kabbaj, Samir |
| Author:
|
Charifi, Ahmed |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
141 |
| Issue:
|
1 |
| Year:
|
2016 |
| Pages:
|
59-70 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\mathcal {H}$ be a complex Hilbert space, $A$ a positive operator with closed range in $\mathscr {B}(\mathcal {H})$ and $\mathscr {B}_{A}(\mathcal {H})$ the sub-algebra of $\mathscr {B}(\mathcal {H})$ of all \mbox {$A$-self}-adjoint operators. Assume $\phi \colon \mathscr {B}_{A}(\mathcal {H})$ onto itself is a linear continuous map. This paper shows that if $\phi $ preserves \mbox {$A$-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 $T \in \mathscr {B}_{A}(\mathcal {H})$, where $P=A^{+}A$ and $A^{+}$ is the Moore-Penrose inverse of $A$. A similar result is also true if $\phi $ preserves \mbox {$A$-quasi}-unitary operators in both directions such that there exists an operator $T$ satisfying $P\phi (T)=P$. (English) |
| Keyword:
|
linear preserver problem |
| Keyword:
|
semi-inner product |
| MSC:
|
15A86 |
| MSC:
|
46C50 |
| idZBL:
|
Zbl 06562158 |
| idMR:
|
MR3475137 |
| DOI:
|
10.21136/MB.2016.4 |
| . |
| Date available:
|
2016-03-17T19:45:46Z |
| Last updated:
|
2020-07-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144851 |
| . |
| Reference:
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