## Annals of Functional Analysis

### A max version of Perron--Frobenius theorem for nonnegative tensor

#### Abstract

In this paper we generalize the max algebra system of nonnegative matrices to the class of nonnegative tensors and derive its fundamental properties. If $\mathbb{A} \in \Re_ + ^{\left[ {m,n} \right]}$ is a nonnegative essentially positive tensor such that satisfies the condition class NC, we prove that there exist $\mu \left( \mathbb{A} \right)$ and a corresponding positive vector $x$ such that $\mathop {\max }\limits_{1 \le{i_2}\cdots {i_m} \le n} \left\{ {{a_{i{i_2}\cdots {i_m}}}{x_{{i_2}}}\cdots {x_{{i_m}}}} \right\}=\mu \left( \mathbb{A} \right) x_i^{m - 1},\,\,\,\,i = 1,2,\cdots ,n.$ This theorem, is well known as the max algebra version of Perron--Frobenius theorem for this new system.

#### Article information

Source
Ann. Funct. Anal., Volume 6, Number 3 (2015), 145-154.

Dates
First available in Project Euclid: 17 April 2015

https://projecteuclid.org/euclid.afa/1429286038

Digital Object Identifier
doi:10.15352/afa/06-3-12

Mathematical Reviews number (MathSciNet)
MR3336911

Zentralblatt MATH identifier
1325.15021

#### Citation

Afshin, Hamid Reza; Shojaeifard, Ali Reza. A max version of Perron--Frobenius theorem for nonnegative tensor. Ann. Funct. Anal. 6 (2015), no. 3, 145--154. doi:10.15352/afa/06-3-12. https://projecteuclid.org/euclid.afa/1429286038

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