Perron-Frobenius theory of seminorms: a topological approach
This interesting and comprehensive paper studies a generalization of the well-known Perron-Frobenius theory. The authors were motivated by the results of \textit{S.~M.~Rump} [Linear Algebra Appl. 266, 1--42 (1997; Zbl 0901.15002), Electron. J. Linear Algebra 9, 112--117 (2002; Zbl 1046.15025); Linear Algebra Appl. 363, 251--273 (2003; Zbl 1019.15012)] who has extended this theory from nonnegative to arbitrary complex matrices by introducing the signed spectral radius \[ \rho^{F}(A)= \max\{\lambda \geq 0 \mid \exists 0\neq x\in F^n\text{ s.t. }| Ax| = \lambda | x| \}. \] In the present paper this notion is further generalized by replacing the linear transformation \(A\) by a vector seminorm, that is a function \(f:F^n\to\mathbb{R}^r\) whose coordinates are seminorms. The theory is further extended from \(F\in\{\mathbb{R},\mathbb{C}\}\) to \(F=\mathbb{R}_p\), the set of all \(p\)-real numbers, i.e.~ \(z\in\mathbb{C}\) such that \(z=| z| \varepsilon\) with \(\varepsilon^p=1\), and moreover to general decompositions of \(F^n\) (thus the results are no more subject to a choice of some special basis). The central part of the paper form a collection of alternative theorems relating inequalities \(f(x) \geq \lambda| x| \) and \(f(x) < \lambda| x| \). These theorems yield a generalized Perron-Frobenius theory for vector seminorms. All the main properties of the original Perron root as well as the results of Rump can be deduced from this general theory. One of the most important properties is the characterization of the spectral radius via the Collatz-Wielandt numbers. Furthermore, also alternative theorems for \(P\)-matrices and strong \(D\)-stable matrices are given. The alternative theorems are obtained as a consequence of topological principles of local and global types. The proofs are involved and use cohomological arguments. Some topological background is provided for readers unfamiliar with these techniques. The paper has a good structure and reads well.
scientific article; zbMATH DE number 2160192
scientific article