The closure property of H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{H}$\end{document}-tensors under the Hadamard product

In this paper, we investigate the closure property of H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {H}$\end{document}-tensors under the Hadamard product. It is shown that the Hadamard products of Hadamard powers of strong H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{H}$\end{document}-tensors are still strong H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{H}$\end{document}-tensors. We then bound the minimal real eigenvalues of the comparison tensors of the Hadamard products involving strong H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{H}$\end{document}-tensors. Finally, we show how to attain the bounds by characterizing these H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{H}$\end{document}-tensors.


Introduction
The study of tensors with their various applications has increasingly attracted extensive attention and interest [-]. A tensor can be regarded as a higher-order generalization of a matrix in linear algebra. However, unlike matrices, the problems for tensors are generally nonlinear. Hence, there is a large need to investigate tensor problems. Recently, some structured tensors such as nonnegative tensors, M-tensors and H-tensors have been introduced and studied well, and many interesting results for these tensors have been ob- An mth-order n-dimensional real tensor A is a multidimensional array of n m real entries of the form A = (a i  ...i m ), a i  ...i m ∈ R,  ≤ i  , . . . , i m ≤ n.
The entries a ii...i are called the diagonal entries of A. If all its off-diagonal entries are zero, then A is diagonal. The identity tensor I is a diagonal tensor all of whose diagonal entries are . In the sequel, we denote by R (m,n) the set of all mth-order n-dimensional real tensors. For a tensor A ∈ R (m,n) and a vector x = (x  , . . . , x n ) T ∈ C n , the tensor-vector multiplication Ax m- is defined as an n-vector whose ith entries are If there are a number λ and a nonzero vector x ∈ C n such that then λ is called the eigenvalue of A and x is the eigenvector of A associated with λ, where Note that the definition of eigenvalues of tensors was independently introduced by Qi [] and Lim []. Denote by ϕ(A) the set of all the eigenvalues of A ∈ R (m,n) , and denote where Re λ is the real part of λ. It is well known that if A ∈ R (m,n) is a nonnegative tensor (i.e., all its entries are nonnegative), then ρ(A) must be its eigenvalue [, ]; and if A ∈ R (m,n) is an M-tensor, then τ (A) must be its eigenvalue [].
A tensor A ∈ R (m,n) is said to be a (strong) M-tensor if A can be written as A = sI -B, where B ∈ R (m,n) is nonnegative and s(>) ≥ ρ(B). In this case, according to the proof of [, Theorem .], τ (A) = sρ(B). For a tensor A = (a i  ...i m ) ∈ R (m,n) , the comparison tensor

For a nonnegative tensor
Many interesting properties have been provided for M-tensors. Recall that A ∈ R (m,n) is an H-tensor if and only if M(A) ∈ R (m,n) is an M-tensor. So using [, Theorem .] and [, Theorem ], we have the following facts on H-tensors that will be frequently used in the next sections: x >  such that M(A)x m- > . Clearly, these interesting results are due to the special structures of H-tensors. So it is natural to consider how to preserve the structure properties under certain operations. In addition, many interesting results have been obtained for the Hadamard products involving M-matrices and H-matrices []. It is natural to ask whether we can provide similar results for the tensor case. Motivated by these facts, the aim of this paper is to investigate the closure property of H-tensors under the Hadamard product.

Definition . Given two tensors
To obtain our results, we need the following two famous inequalities: • Hölder's inequality: let a i and b i be nonnegative numbers for i = , , . . . , n, and let  < r < . Then . . , a n . The rest of the paper is organized as follows. In Section , we show the closure property of the Hadamard products of Hadamard powers of strong H-tensors. In Section , we bound the minimal real eigenvalues of the comparison tensors of the Hadamard products involving strong H-tensors. In Section , we characterize these strong H-tensors such that the bounds can be obtained.

The closure property
In this section, we provide the closure property of the Hadamard products of Hadamard powers of strong H-tensors.

Lemma . Let A, B ∈ R (m,n) be strong H-tensors and let
respectively. This means that, for all i = , , . . . , n, Note that  ≤ r ≤ . Thus, using the Hölder inequality, we have Clearly, there exists a positive vector x = (x i ) ∈ R n such that M(A)x m- >  and so, for all i = , , . . . , n, from which we get, by considering t ≥  and using the Minkowski inequality, is a strong H-tensor by (P). The result is proved. Now we are ready to present the main result of this section. Theorem . Let A  , . . . , A k ∈ R (m,n) be strong H-tensors and let r  , . . . , r k be positive num- Proof Consider that A ∈ R (m,n) is a strong H-tensor if and only if |A| ∈ R (m,n) is a strong H-tensor. So, without loss of generality, assume that all the tensors A i are nonnegative for i = , , . . . , k. We first use the induction on k to prove the result in the case that k i= r i = . Clearly, the result is true for k =  by Lemma .. Assume that the result is true for k -. Now let Recall that each A i is nonnegative. Then is a strong H-tensor. The result is proved.
be defined as follows: we see by (P) that D is a strong H-tensor.

Bounding the minimal real eigenvalues
In this section, we bound the minimal real eigenvalues of the comparison tensors of the Hadamard products involving strong H-tensors.
The result is proved.

Lemma . Let
Then, by the Minkowski inequality, we have, for all i = , , . . . , n, The result is proved.
Our main result of this section is the following.
Theorem . Let A  , A  , . . . , A k ∈ R (m,n) be strong H-tensors and let r  , r  , . . . , r k be positive numbers such that k i= r i ≥ . Then Proof By (P), without loss of generality, assume that all the tensors A i are nonnegative for i = , , . . . , k. We first use the induction on k to prove the result in the case that k i= r i = . Obviously, the result is true for k =  by Lemma .. Assume the result is true for k -. Now let Consider that each A i is nonnegative. Then Note that k- i= r i -r k = . Thus B is a strong H-tensor by Theorem .. Therefore, using the induction assumption, we get So the result is true in the case that k i= r i = . Now we consider the general case t = k i= r i ≥ . Set l i = r i t - for i = , , . . . , k. Then k is a strong H-tensor by Theorem .. Therefore, according to the case above, using Lemma . we get The result is proved.

Characterizations for the equality case
In this section, we characterize the strong H-tensors such that the equality of (.) holds. For a tensor A = (a i  i  ...i m ) ∈ R (m,n) and a nonsingular diagonal matrix It must be pointed out that This means that the two Hölder inequalities of (.) are equalities and so, for all i = , , . . . , n, By considering (.), Therefore we have, for all i = , , . . . , n, . . , y n x n ) ∈ R n×n and γ = σ (A) σ (B) . Then (.) implies that |A| = γ |B|D -(m-) · D · · · D m- . The result is proved. Now we characterize strong H-tensors such that the equality of (.) holds in the case that k i= r i = .
Theorem . Let A  , A  , . . . , A k ∈ R (m,n) be strong H-tensors and let r  , r  , . . . , r k be positive numbers such that k i= r i = . Then where γ i >  and D i ∈ R n×n is a positive diagonal matrix.
• Using (.) and (.), we derive that there exist γ k >  and a positive diagonal matrix D k ∈ R n×n such that Thus the result is proved.
Next we characterize strong H-tensors such that the equality of (.) holds in the case that k i= r i > .