Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

Query:
Fill in the form and click »Search«...
Format:
Display: entries per page entries
Zbl 1185.47037
Yang, Li; Zhang, Ling
Maps on ${\cal B}({\cal H})$ preserving involution.
(English)
[J] Linear Algebra Appl. 431, No. 5-7, 666-672 (2009). ISSN 0024-3795

This paper belongs to a recent spate of papers (see, for example, [Linear Algebra Appl.\ 431, No.\,5--7, 833--842 (2009; Zbl 1183.47031); ibid., 974--984 (2009; Zbl 1183.15017)] in the same issue as the paper being reviewed, and references therein). The common thread of these papers is to characterize maps on Hilbert space satisfying a certain property (preservers), without assuming linearity. For instance, in the paper under review, the following is proven: Given an infinite-dimensional Hilbert space ${\cal H}$, let $\Gamma=\{A\in{\cal B}({\cal H}): A^2= \text{id}_{\cal H}\}$, and let $\varphi:{\cal B}({\cal H})\to{\cal B}({\cal H})$, such that $$A-\lambda B\in\Gamma\iff \varphi(A)-\lambda\varphi(B)\in\Gamma\text { for all }A,B\in{\cal B}({\cal H}), \quad \lambda\in\Bbb C.$$ Then either: {\parindent=7mm \item{(i)} $\varphi(A)=\pm TAT^{-1}$, with $T\in\text{GL}({\cal H})$, or \item{(ii)} $\varphi(A)=\pm TA^*T^{-1}$, with $T$ invertible and conjugate linear. \par}
[Mart\'in Argerami (Regina)]
MSC 2000:
*47B49 Transformers (=operators on spaces of operators)
15A04 Linear transformations (linear algebra)

Keywords: symmetry; involution; Hilbert space; preserver problem

Citations: Zbl 1183.47031; Zbl 1183.15017

Highlights
Master Server

### Zentralblatt MATH Berlin [Germany]

© FIZ Karlsruhe GmbH

Zentralblatt MATH master server is maintained by the Editorial Office in Berlin, Section Mathematics and Computer Science of FIZ Karlsruhe and is updated daily.

Other Mirror Sites

Copyright © 2013 Zentralblatt MATH | European Mathematical Society | FIZ Karlsruhe | Heidelberg Academy of Sciences