Some inequalities on the spectral radius of nonnegative tensors

Abstract The eigenvalues and the spectral radius of nonnegative tensors have been extensively studied in recent years. In this paper, we investigate the analytic properties of nonnegative tensors and give some inequalities on the spectral radius.

In the proof of Theorem 2.3 in [23], the authors considered the sequence of nonnegative tensors and gave the limit formula regarding the spectral radius. Note that the result holds when the sequence is monotonic. We want to know whether the result still holds when the sequence is not monotonic and try to investigate the continuity of the spectral radius of nonnegative tensors by some inequalities. Recently, Sun et al. generalized some inequalities on the spectral radius of the Hadamard product of nonnegative matrices to nonnegative tensors [29]. Their beautiful results make us interested in the further study of the Hadamard product of tensors.
In this paper, we mainly investigate the analytic properties of the spectral radius of nonnegative tensors. We discuss the continuity of the spectral radius by means of limit formulas as well as tensor inequalities involving norms. We also give some inequalities on the spectral radius of the Hadamard product of nonnegative tensors. These results can be seen as a generalization of the existing inequalities on the spectral radius of nonnegative matrices.
The paper is organized as follows. In Section 2, we collect some definitions, notations and helpful lemmas. In Section 3, we discuss the continuity of the spectral radius. In Section 4, we give some inequalities on the spectral radius involving the Hadamard product.

Preliminaries
is a multiarray of real entries ⋯ a , where i j ∈ {1, 2,…,n} for j ∈ {1, 2,…,m}. When m = 2, is a matrix of order n. The set of all mth order n-dimensional real tensors is denoted as T m,n . Throughout this paper, we assume that m, n ≥ 2.
A tensor is said to be nonnegative (positive) if each of its entry is nonnegative (positive). Denote by the zero tensor, and by the tensor with each entry equal to 1. For a tensor , ≥ (>) implies that is nonnegative (positive). For two tensors and , ≥ or ≤ implies that − is nonnegative. Let | | be the tensor obtained from by taking the absolute values of the entries. Then, | | is nonnegative and denoted by 〈 〉 , , is defined as follows: The Frobenius norm of is defined and denoted as || || = 〈 〉 , for i = 1, 2,…,n. Denote by 0 the zero vector. A complex number λ is called an eigenvalue of if it together with ≠ x 0 forms a solution to the following system of homogeneous polynomial equations: It is well known that the Perron-Frobenius theorem is a fundamental result for nonnegative matrices [30, p. 123]. Chang, Pearson and Zhang generalized this theorem to nonnegative tensors. Yang and Yang gave some further results on the Perron-Frobenius theorem for nonnegative tensors. We summarize some of their results as follows.

Continuity of the spectral radius of nonnegative tensors
If has an eigenvalue with a positive eigenvector corresponding to it, then   Proof.
Since the function f(t) = log t is concave on (0, +∞), we have  Thus,     Similarly, for two vectors x = (x 1 , x 2 ,…,x n ) T and y = (y 1 , y 2 ,…,y n ) T , let ∘ = ( … ) x y x y x y x y , , , ; Proof. By continuity of the spectral radius, we may suppose By Lemma 2.2, for i = 1, 2,…,k, there exist  In this paper, we focus on the analytic properties of the spectral radius of nonnegative tensors. First, we discuss the continuity of the spectral radius. Then, we give some inequalities on the spectral radius involving the Hadamard product. These results generalize some existing results on the spectral properties of nonnegative matrices to nonnegative tensors.