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**Hilbert’s projective metric and iterated nonlinear maps.**
*(English)*
Zbl 0666.47028

Mem. Am. Math. Soc. 391, 137 p. (1988).

This two parts work is actually a complete version of author’s announced studies [Proc. NATO Adv. Study Inst., Lisbon/Port. 1986, NATO ASI Ser., Ser. F 37, 231-248 (1987; Zbl 0643.47050)]. From author’s abstract:

“Let K be a cone with nonempty interior \(\dot K\) in a Banach space X and f: \(\dot K\rightarrowtail \dot K\) a map. This paper treats questions like the following: Does f have a suitable normalized eigenvector u in the interior of K? Is the normalized eigenvector u unique? If \(f(tu)=tu\) for every \(t>0\), is it true that for every \(x\in \dot K\) there exists \(\lambda (x)>0\) such that \(\lim_{k\to \infty}f^ k(x)=\lambda (x)u?\) What can be said about the structure of the eigenvectors of f in \(\dot K\)? The class of maps studied includes maps which are homogeneous of degree one and perserve the partial ordering induced by K. Applications are made to the theory of means and their iterates.”

The introduction specifies the above problems while the chapters are successively devoted to basic properties of Hilbert’s projective metric, uniqueness and global stability for eigenvectors, iterates of unnormalized maps, nonexpansive maps and Hilbert’s projective metric.

“Let K be a cone with nonempty interior \(\dot K\) in a Banach space X and f: \(\dot K\rightarrowtail \dot K\) a map. This paper treats questions like the following: Does f have a suitable normalized eigenvector u in the interior of K? Is the normalized eigenvector u unique? If \(f(tu)=tu\) for every \(t>0\), is it true that for every \(x\in \dot K\) there exists \(\lambda (x)>0\) such that \(\lim_{k\to \infty}f^ k(x)=\lambda (x)u?\) What can be said about the structure of the eigenvectors of f in \(\dot K\)? The class of maps studied includes maps which are homogeneous of degree one and perserve the partial ordering induced by K. Applications are made to the theory of means and their iterates.”

The introduction specifies the above problems while the chapters are successively devoted to basic properties of Hilbert’s projective metric, uniqueness and global stability for eigenvectors, iterates of unnormalized maps, nonexpansive maps and Hilbert’s projective metric.

Reviewer: D.Pascali

### MSC:

47H07 | Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces |

47H10 | Fixed-point theorems |

47J10 | Nonlinear spectral theory, nonlinear eigenvalue problems |

47B60 | Linear operators on ordered spaces |