Eigenvalues and Eigenfunctionals of Diagonally Dominant Endomorphisms in Min-Max Analysis
Résumé
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 $f$ to produce $Af$, where, $\forall x$:
$$Af(x) = \min_{y}\{\max\{A(x,y);f(y)\}\}.$$
We present here a complete characterization of eigenvalues and eigen-functionals of diagonally dominant endomorphisms (i.e. such that $\forall x$, $\forall y$: $A(x, x) = 0_A$, $A(x, y) \geq 0_A$). It is shown, in particular, that any real value $\lambda > 0_A$ is an eigenvalue, and that the associated eigen-semi-module has a unique minimal generator.