Stability and paracontractivity of discrete linear inclusions (Q1573666)

A square matrix \(A\) is called paracontracting (PC) with respect to a given norm if \(Ax\neq x\Rightarrow \|Ax\|<\|x\|\). A set \(\Sigma\) of matrices is PC if all elements of \(\Sigma\) are PC with respect to the same norm. The main result of this paper is the proof of a conjecture [cf. \textit{W.-J. Beyn} and \textit{L. Elsner}, Electron. J. Linear Algebra 2, 1-8 (1997; Zbl 0893.65019)] that a finite set \(\Sigma\) of \(n\times n\) matrices is PC iff each infinite trajectory \(x_{j+1}=A_jx_j\) (\(A_j\in\Sigma\)) converges to a point \(x^*\in \mathbb R^n\), i.e. all infinite products of elements of \(\Sigma\) are convergent. Moreover, it is shown that this is equivalent to \(\lim_{j\to\infty}\|x_{j+1}-x_j\|=0\) and also to the fact that \(\Sigma\) is product bounded, i.e. \(\exists C>0\) such that \(\|A_1\cdots A_m\|<C\) for all finite products of (not necessarily different) elements of \(\Sigma\).