On linear preservers of normal maps
For a closed subgroup \(G\) of an orthogonal group on a real finite-dimensional vector space \(V\), an order related to \(G\) is defined by \(y\leq_G x\) if \(y\) is in the convex hull of \(Gx=\{gx:g \in G \}\). The order is said to be a group induced cone (GIC) if there exists a closed convex cone \(D\) with \(D\cap Gx\) non-empty for all \(x\in V\) and \(\langle x,gy\rangle\leq\langle x,y\rangle\) \((x,y\in D\), \(g\in G)\). The map assigning \(x\) the element \(D\cap Gx\) (one element) is called the normal map. For example, the eigenvalue map on the Hermitian matrices (\(n\times n\)) can be viewed as a normal map. For two linear spaces \(V\) and \( W\) together with associated groups and cones, a map \(K\) from \(V\) to \(W\) is called a preserver of normal maps if the normal map of \(Kx\) is equal to \(K\) of the normal map of \(x\). The paper studies preservers of normal maps and properties maintained under the action of these preservers. It is shown, for example, that preservers of normal maps are positive operators preserving the order (induced by the subgroups) with the image of the cone of \(V\) a subset of the cone of \(W\). Numerous examples are provided.
scientific article; zbMATH DE number 5190767