Eigenvalues and eigen-functionals of diagonally dominant endomorphisms in Min-Max analysis

摘要

The so-called (Min, +) analysis may be viewed as an extension to the continuous case and to functional spaces of shortest path algebras in graphs. We investigate here (Min-Max) analysis which extends, in some similar way, minimum spanning tree problems and maximum capacity path problems in graphs. An endomorphism A of the functional Min-Max semi-module acts on any functional f to produce Af, where, For All x: [GRAPHICS] We present here a complete characterization of eigenvalues and eigen-functionals of diagonally dominant endomorphisms (i.e. such that For All x, For All y: A(x, x) = theta(A), A(x,y) greater than or equal to theta(A)). It is shown, in particular, that any real value lambda > theta(A) is an eigenvalue, and that the associated eigen-semi-module has a unique minimal generator. (C) 1998 Published by Elsevier Science Inc. All rights reserved. [References: 17]