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

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 endomorphisms A of the functional Min-Max semi-module acts on any functional ƒ to produce Aƒ, where, for allχ: image We present here a complete characterization of eigenvalues and eigen-functionals of diagonally dominant endomorphisms (i.e. such that for allx, for ally: A(x, x) = 0A, A(x, y) greater-or-equal, slanted 0A). It is shown, in particular, that any real value λ > 0A is an eigenvalue, and that the associated eigen-semi-module has a unique minimal generator.