The Perron-Frobenius theorem and the Hilbert projective distance (Q2731658)

In the notation of the paper, let \(V\) be a cone in a real vector space \(D.\) Assume that \(\rho_V(x,y)=\sup\{\lambda:x-\lambda y \in V\}<\infty\) for all \(x,y\in V\setminus \{0\}.\) Then the projective distance in \(V\) is defined by \(\varphi_V(x,y)=\rho_V(x,y)\rho_V(y,x).\) (\(-\log\varphi_V(x,y)\) defines a pseudometric.) This notion, explained at length in section 2, allows to define (as do others) the convergence of a sequence of semi-rays with origin in 0 towards a given one; i.e. of `directional convergence'.NEWLINENEWLINENEWLINEThe author studies for a \(u\in V=\mathbb R^{n}_+\) the directional convergence of the sequence \(\{B^k u\}_{k\in \mathbb N},\) \(B\) an entrywise positive matrix, and bases on it an intuitively appealing though not particularly short proof of the well known Perron Frobenius theorem for irreducible nonnegative matrices, `different from text-book proofs'; a claim that the reviewer found possibly true.