多齐次映射的Perron-Frobenius定理
The Perron-Frobenius theorem for multi-homogeneous mappings
摘要：
非负矩阵的Perron-Frobenius理论已推广到锥上的保序齐次映射，最近又推广到非负多重线性型。我们统一这两种方法通过引入保序多齐次映射的概念，其相关的非线性谱问题和谱半径。我们显示了几个Perron-Frobenius型结果，这些映射解决的存在性，唯一性和极大值的非负和正特征对。我们证明了一个Collatz-Wielandt原理和谱半径的其他特征，并分析了这些映射的迭代收敛到其唯一的正特征向量。在提供一个新的扩展的非线性Perron-Frobenius理论的多维情况下，我们的贡献构成了几个改进和更深入地了解当前的谱理论的非负张量的基础。事实上，近年来，重要的结果已经通过重铸某些谱方程的多重线性型的齐次映射，然而，我们的方法是更适合这样的问题，这些结果可以进一步细化和改进，通过采用我们的新的多齐次设置。
英文摘要：
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 multi-homogeneous 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 towards their unique positive eigenvectors. On top of providing a new extension of the nonlinear Perron-Frobenius theory to the multi-dimensional 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 multi-homogeneous setting.
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