When do several linear operators share an invariant cone?
Consider a finite family of matrices \(\mathcal A =\{A_1,A_2,\dots,A_m\},~m \geq 1\) where each \(A_i \in \mathbb{R}^{d \times d}\). A subset \(K\) of \(\mathbb{R}^d\) is called a cone if \(K\) is closed, has nonempty interior, does not contain a straight line and has a vertex at the origin. A cone \(K\) is called invariant for \(\mathcal A\), if \(A_i(K) \subseteq K\) for all \(i=1,2,\dots,m\). For the class \(\mathcal A\), define the ``average`` matrix \(\bar{A}=\frac{1}{m}\sum_{i=1}^m~A_i\) and the \(L_1\)-spectral radius of \(\mathcal A\) by \[ \rho (\mathcal A) = \lim_{k \rightarrow \infty} \left(\frac{1}{m^k} \sum_{d_1,d_2\dots,d_k \in \{1,2,\dots,m\}}~\| A_{d_1}.A_{d_2}\dots A_{d_k} \|\right)^{\frac{1}{k}}. \] For any matrix \(B\) having the property that it has an eigenvalue equal to its spectral radius, we denote by \(\lambda_{\max}(B)\) its largest eigenvalue. The main result (Theorem 1) of the paper under review is that if \(\mathcal A\) is an irreducible family, then \(\mathcal A\) has an invariant cone if and only if \(\rho (\mathcal A) =\lambda_{\max}(\bar{A})\). Easily verifiable sufficient and necessary conditions for the existence of an invariant cone are derived from this result. The author then analyzes the algorithmic complexity of the problem of determining whether a family of matrices has an invariant cone, and formulates an approximative analogue by introducing the \textit{co-directional number} for an irreducible family of matrices. An algorithm for computing this is presented. Several open problems are proposed.
scientific article; zbMATH DE number 5726561