CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCMB
Publications        
Abstract view

Linear Maps on Selfadjoint Operators Preserving Invertibility, Positive Definiteness, Numerical Range

  Published:2003-06-01
 Printed: Jun 2003
  • Chi-Kwong Li
  • Leiba Rodman
  • Peter Šemrl
Features coming soon:
Citations   (via CrossRef) Tools: Search Google Scholar:
Format:   HTML   LaTeX   MathJax   PDF   PostScript  

Abstract

Let $H$ be a complex Hilbert space, and $\HH$ be the real linear space of bounded selfadjoint operators on $H$. We study linear maps $\phi\colon \HH \to \HH$ leaving invariant various properties such as invertibility, positive definiteness, numerical range, {\it etc}. The maps $\phi$ are not assumed {\it a priori\/} continuous. It is shown that under an appropriate surjective or injective assumption $\phi$ has the form $X \mapsto \xi TXT^*$ or $X \mapsto \xi TX^tT^*$, for a suitable invertible or unitary $T$ and $\xi\in\{1, -1\}$, where $X^t$ stands for the transpose of $X$ relative to some orthonormal basis. Examples are given to show that the surjective or injective assumption cannot be relaxed. The results are extended to complex linear maps on the algebra of bounded linear operators on $H$. Similar results are proved for the (real) linear space of (selfadjoint) operators of the form $\alpha I+K$, where $\alpha$ is a scalar and $K$ is compact.
Keywords: linear map, selfadjoint operator, invertible, positive definite, numerical range linear map, selfadjoint operator, invertible, positive definite, numerical range
MSC Classifications: 47B15, 47B49 show english descriptions Hermitian and normal operators (spectral measures, functional calculus, etc.)
Transformers, preservers (operators on spaces of operators)
47B15 - Hermitian and normal operators (spectral measures, functional calculus, etc.)
47B49 - Transformers, preservers (operators on spaces of operators)
 

© Canadian Mathematical Society, 2014 : https://cms.math.ca/