Positivity in complex spaces and its application to Gershgorin discs (Q799755)

For a given set S or row vectors \(S=\{y^ H_ 1,...,y^ H_ m\}\), span \(S={\mathbb{C}}^ n\) let \(G_ i(A)=\{y^ H_ iAx| y^ H_ ix=1,\quad| y^ H_ jx| |\leq 1,\quad j=1,...,m\} i=1,...,m\), A \(n\times n\)-matrix, and \(\tilde G_ i(A)=\{y^ H_ iAx| y^ H_ jx|\leq 1,j=1,...,m\}.\) The main result is Theorem 5: If \(\tilde G_ i(A)\cap G_ j(A)=\emptyset\), for all j, \(j\neq i\), then \(G_ i(A)\) contains exactly one eigenvalue, and this eigenvalue is simple. From this follows (i) the well known result on separated Gerschgorin disks and (ii) the result that if \(G_ i\) is separated from the convex hull of \(\cup\{G_ j,j\neq i\}\), then \(G_ i\) contains a simple eigenvalue. For the proof the author establishes a generalization of some of the Krein-Rutman results on existence of eigenvectors in K belonging to eigenvalues with maximal modulus to weakly nonnegative and weakly positive mappings A (i.e., \(AK\subseteq K^ C\) and \(A(K\backslash\{0\})\subseteq int(K^ C)\) resp., where K is a cone in a vector space over F, where \(F={\mathbb{R}}\) or \({\mathbb{C}}\), and \(K^ C=\{\sigma x: x\in K,\quad\sigma \in F\}\) its circular hull).