The Perron--Frobenius Theorem for Multihomogeneous Mappings

Abstract

The Perron--Frobenius theory for nonnegative matrices has been generalized to order-preserving homogeneous mappings on a cone and more recently to nonnegative multilinear forms. We unify both approaches by introducing the concept of order-preserving multihomogeneous mappings, their associated nonlinear spectral problems, and spectral radii. We show several Perron--Frobenius type results for these mappings addressing existence, uniqueness, and maximality of nonnegative and positive eigenpairs. We prove a Collatz--Wielandt principle and other characterizations of the spectral radius and analyze the convergence of iterates of these mappings toward their unique positive eigenvectors. On top of providing a new extension of the nonlinear Perron--Frobenius theory to the multidimensional case, our contribution poses the basis for several improvements and a deeper understanding of the current spectral theory for nonnegative tensors. In fact, in recent years, important results have been obtained by recasting certain spectral equations for multilinear forms in terms of homogeneous maps; however, as our approach is more adapted to such problems, these results can be further refined and improved by employing our new multihomogeneous setting.