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

Published online by Cambridge University Press:  20 November 2018

Chi-Kwong Li ,
Leiba Rodman  and
Peter Šemrl
Chi-Kwong Li
Affiliation:
Department of Mathematics, College of William and Mary, P.O. Box 8795, Williamsburg, Virginia 23187-8795, U.S.A., e-mail: ckli@math.wm.edu
Leiba Rodman
Affiliation:
Department of Mathematics, College of William and Mary, P.O. Box 8795, Williamsburg, Virginia 23187-8795, U.S.A., e-mail: lxrodm@math.wm.edu
Peter Šemrl
Affiliation:
Department of Mathematics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia, e-mail: peter.semrl@fmf.uni-lj.si
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Abstract

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Let $H$ be a complex Hilbert space, and $\mathcal{H}\left( H \right)$ be the real linear space of bounded selfadjoint operators on $H$. We study linear maps $\phi :\,\mathcal{H}(H)\,\to \,\mathcal{H}(H)$ leaving invariant various properties such as invertibility, positive definiteness, numerical range, etc. The maps $\phi$ are not assumed a priori continuous. It is shown that under an appropriate surjective or injective assumption $\phi$ has the form $X\mapsto \xi TX{{T}^{*}}orX\mapsto \xi T{{X}^{t}}{{T}^{*}}$, 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

47B1547B49linear mapselfadjoint operatorinvertiblepositive definitenumerical range
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 46 , Issue 2 , 01 June 2003 , pp. 216 - 228
Copyright
Copyright © Canadian Mathematical Society 2003

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