Let \(M_{n}({\mathbb {R}}_{+})\) be the set of all \(n \times n\) nonnegative matrices. Recently, in Tavakolipour and Shakeri (Linear Multilinear Algebra 67, 2019, https://doi.org/10.1080/03081087.2018.1478946), the concept of the numerical range in tropical algebra was introduced and an explicit formula describing it was obtained. We study the isomorphic notion of the numerical range of nonnegative matrices in max algebra and give a short proof of the known formula. Moreover, we study several generalizations of the numerical range in max algebra. Let \(1 \le k \le n\) be a positive integer and \(C \in M_{ n}({\mathbb {R}}_{+}).\) We introduce the notions of max \(k-\)numerical range and max \(C-\)numerical range. Some algebraic and geometric properties of them are investigated. Also, max numerical range \(W_\text {max}(\varSigma )\) of a bounded set \(\varSigma\) of \(n \times n\) nonnegative matrices is introduced and some of its properties are also investigated.

中文翻译：

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关于最大代数矩阵的数值范围

令\（M_ {n}（{\ mathbb {R}} _ {+}）\）为所有\（n×n \）个非负矩阵的集合。最近，在Tavakolipour和Shakeri（线性多线性代数67，2019，https://doi.org/10.1080/03081087.2018.1478946）中，引入了热带代数中数值范围的概念，并获得了描述它的显式公式。我们研究了最大代数中非负矩阵数值范围的同构概念，并给出了已知公式的简短证明。此外，我们研究了最大代数中数值范围的几种概括。令\（1 \ le k \ le n \）为正整数，令\（C \ in M_ {n}（{\ mathbb {R}} _ {+}）。\）我们介绍max \（ k- \）数值范围和最大值\（C- \）数值范围。研究了它们的一些代数和几何性质。另外，介绍了非负矩阵\（n \ times n \）的有界集合\（\ varSigma \）的最大数值范围\（W_ \ text {max}（\ varSigma）\）并研究了其某些性质。