Bounds for the Z-spectral radius of nonnegative tensors

In this paper, we have proposed some new upper bounds for the largest Z-eigenvalue of an irreducible weakly symmetric and nonnegative tensor, which improve the known upper bounds obtained in Chang et al. (Linear Algebra Appl 438:4166–4182, 2013), Song and Qi (SIAM J Matrix Anal Appl 34:1581–1595, 2013), He and Huang (Appl Math Lett 38:110–114, 2014), Li et al. (J Comput Anal Appl 483:182–199, 2015), He (J Comput Anal Appl 20:1290–1301, 2016).


Definition 1
The square tensor A is called reducible if there exists a nonempty proper index subset J ⊂ {1, 2, . . . , n} such that a i 1 ,i 2 ,...,i m = 0, ∀i 1 ∈ J, ∀i 2 , . . . , i m / ∈ J. If A is not reducible, then we call A to be irreducible.

Definition 2 Let
A be an m-order and n-dimensional tensor. We define σ (A) the Z-spectrum of A by the set of all Z-eigenvalues of A. Assume σ (A) � = ∅, then the Z-spectral radius of A is denoted by Let N = {1, 2, . . . , n}. In 2013, Chang et al. gave the following bound for the Z-eigenvalues of an m-order n-dimensional tensor A.

Theorem 1 Let A be an m-order and n-dimensional tensor. Then
For the positively homogeneous operators, Song and Qi (2013) studied the relationship between the Gelfand formula and the spectral radius as well as the upper bound of the spectral radius. From Corollary 4.5 in Song and Qi (2013), we can get the following result:

Theorem 2 Let A be an m-order and n-dimensional tensor. Then
We shall denote the set of all mth order n dimensional tensors by R [m,n] , and the set of all nonnegative (or, respectively, positive) mth order n dimensional tensors by R [m,n] + (or, respectively, R [m,n] ++ ). If the tensor is positive, He and Huang gave the following Z-eigenpair bound (see Theorem 2.7 of He and Huang 2014):  He et al. SpringerPlus (2016) 5:1727 where A real tensor of order m dimension n is called the unit tensor, if its entries are δ i 1 ...i m for i 1 , . . . , i m ∈ N, where And we define He gave the following upper bound (see Theorem 3.3 of He 2016): is an irreducible weakly symmetric tensor. Then where Our goal in this paper is to show some tighter upper bounds for the largest Z-eigenvalue of a nonnegative tensor. In section "Main results", some new upper bounds for the largest Z-eigenvalue are obtained, which are tighter than the results in Theorems 1-5 (Chang et al. 2013;Song and Qi 2013;He and Huang 2014;He 2016).

Main results
In this section, we consider some new upper bounds for the largest Z-eigenvalue of a nonnegative tensor.
A tensor A is called weakly symmetric if the associated homogeneous polynomial Ax m satisfies This concept was first introduced and used by Chang et al. (2013) for studying the properties of Z-eigenvalue of nonnegative tensors and presented the following Perron-Frobenius Theorem for the Z-eigenvalue of nonnegative tensors. ∇Ax m = mAx m−1 .

Lemma 1 Suppose that
is an irreducible weakly symmetric tensor, then the spectral radius ρ(A) is a positive Z-eigenvalue with a positive Z-eigenvector.
Based on the lemma, we give our main results as follows.  (7) and (8), we obtain, we obtain (9)

Therefore
This must be true for every s � = t, then, we get And this could be true for any t ∈ N, that is Thus, we complete the proof.

Remark 1 Obviously, we can get
That is to say, the bound in Theorem 6 is always better than the result in Theorem 5. We denote And let

Theorem 7 Suppose that
is an irreducible weakly symmetric tensor. Then where Proof First, Let x = (x 1 , . . . , x n ) T be an Z-eigenvector of A corresponding to ρ(A), that is, i,j (A) . He et al. SpringerPlus (2016) 5:1727 Remark 2 Let i be a nonempty proper subset of i , we have that for (i 2 , . . . , i m ) ∈ � i , Similar to the proof of Theorem 7, we can get where which is always better than the result in Theorem 6.
Example 1 We now show the efficiency of the new upper bounds in Theorems 6 and 7 by the following example. Consider the tensor A = (a ijk ) and of order 3 dimension 3 with entries defined as follows: By This example shows that the bound in Theorem 7 is the best among the known bounds.