Linear maps preserving -unitary operators
Abdellatif Chahbi; Samir Kabbaj; Ahmed Charifi
Mathematica Bohemica (2016)
- Volume: 141, Issue: 1, page 59-70
- ISSN: 0862-7959
Access Full Article
top
Abstract
topHow to cite
topChahbi, 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 -
References
top- Arias, M. L., Corach, G., Gonzalez, M. C., 10.1007/s00020-008-1613-6, Integral Equations Oper. Theory 62 (2008), 11-28. (2008) Zbl1181.46018MR2442900DOI10.1007/s00020-008-1613-6
- Arias, M. L., Corach, G., Gonzalez, M. C., Partial isometries in semi-Hilbertian spaces, Linear Algebra Appl. 428 (2008), 1460-1475. (2008) Zbl1140.46009MR2388631
- Brešar, M., {Š}emrl, P., 10.1006/jfan.1996.0153, J. Funct. Anal. 142 (1996), Article No. 0153, 360-368. (1996) Zbl0873.47002MR1423038DOI10.1006/jfan.1996.0153
- Brešar, M., {Š}emrl, P., 10.4153/CJM-1993-025-4, Can. J. Math. 45 (1993), 483-496. (1993) Zbl0796.15001MR1222512DOI10.4153/CJM-1993-025-4
- Douglas, R. G., On the operator equation and related topics, Acta Sci. Math. 30 (1969), 19-32. (1969) MR0250106
- Douglas, R. G., 10.1090/S0002-9939-1966-0203464-1, Proc. Am. Math. Soc. 17 (1966), 413-415. (1966) Zbl0146.12503MR0203464DOI10.1090/S0002-9939-1966-0203464-1
- Frobenius, G., Über die Darstellung der endlichen Gruppen durch lineare Substitutionen, Berl. Ber. 1897 German (1897), 994-1015. (1897)
- Herstein, I. N., Topics in Ring Theory, Chicago Lectures in Mathematics The University of Chicago Press, Chicago (1969). (1969) Zbl0232.16001MR0271135
- Li, C.-K., Tsing, N.-K., Linear preserver problems: A brief introduction and some special techniques, Linear Algebra Appl. 162-164 (1992), 217-235. (1992) Zbl0762.15016MR1148401
- Mbekhta, M., Linear maps preserving the generalized spectrum, Extr. Math. 22 (2007), 45-54. (2007) Zbl1160.47033MR2368700
- Omladič, M., {Š}emrl, P., 10.1090/S0002-9939-1995-1254849-4, Proc. Am. Math. Soc. 123 (1995), 1069-1074. (1995) Zbl0831.47026MR1254849DOI10.1090/S0002-9939-1995-1254849-4
- Palmer, T. W., Banach Algebras and the General Theory of -algebras, II, Encyclopedia of Mathematics and Its Applications 79 Cambridge University Press, Cambridge (2001). (2001) MR1819503
- Palmer, T. W., -representations of -algebras, Proc. Int. Symp. on Operator Theory (Indiana Univ., Bloomington, Ind., 1970) 20 Cambridge University Press, Cambridge (2001), 929-933. (2001) MR0410396
- Phadke, S. V., Khasbardar, S. K., Thakare, N. K., On QU-operators, Indian J. Pure Appl. Math. 8 (1977), 335-343. (1977) Zbl0367.47015MR0463961
- Pierce, S., General introduction: A survey of linear preserver problems, Linear and Multilinear Algebra 33 (1992), 3-5. (1992) MR1346778
- Pt{á}k, V., 10.1007/BF01304613, Manuscr. Math. 6 (1972), 245-290. (1972) Zbl0304.46036MR0296705DOI10.1007/BF01304613
- Russo, B., Dye, H. A., 10.1215/S0012-7094-66-03346-1, Duke Math. J. 33 (1966), 413-416. (1966) MR0193530DOI10.1215/S0012-7094-66-03346-1
- {Š}emrl, P., 10.4064/sm-105-2-143-149, Stud. Math. 105 (1993), 143-149. (1993) Zbl0810.47001MR1226624DOI10.4064/sm-105-2-143-149
- Watkins, W., 10.1016/0024-3795(76)90060-4, Linear Algebra Appl. 14 (1976), 29-35. (1976) Zbl0329.15005MR0480574DOI10.1016/0024-3795(76)90060-4
NotesEmbed ?
topYou must be logged in to post comments.
To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.