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

[摘要] Let $H$ be a complex Hilbert space, and $HH$ be the real linear space ofbounded selfadjoint operators on $H$. We study linear maps $phicolon HHo HH$ leaving invariant various properties such as invertibility, positivedefiniteness, numerical range, {it etc}. The maps $phi$ are not assumed{it a priori/} continuous. It is shown that under an appropriate surjectiveor injective assumption $phi$ has the form $X mapsto xi TXT^*$ or $X mapstoxi TX^tT^*$, for a suitable invertible or unitary $T$ and $xiin{1, -1}$,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 cannotbe relaxed. The results are extended to complex linear maps on the algebra ofbounded 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.

[学科分类] 数学（综合）

[关键词] linear map;selfadjoint operator;invertible;positive definite;numerical range