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Infinite and finite dimensional generalized Hilbert tensors
Linear Algebra and its Applications 532 (2017) 8–24
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Linear Algebra and its Applications www.elsevier.com/locate/laa
Infinite and finite dimensional generalized Hilbert tensors ✩ Wei Mei a , Yisheng Song b,∗ a
School of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan, 453007, PR China b School of Mathematics and Information Science and Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control, Henan Normal University, Xinxiang, Henan, 453007, PR China
a r t i c l e
i n f o
Article history: Received 16 November 2016 Accepted 11 May 2017 Available online 23 June 2017 Submitted by R. Brualdi MSC: 47H15 47H12 34B10 47A52 47J10 47H09 15A48 47H07 Keywords: Generalized Hilbert tensor Spectral radius Norm Upper bound
a b s t r a c t In this paper, we introduce the concept of an m-order n-dimensional generalized Hilbert tensor Hn = (Hi1 i2 ···im ), Hi1 i2 ···im =
1 , i1 + i2 + · · · i m − m + a
a ∈ R \ Z− ; i1 , i2 , · · · , im = 1, 2, · · · , n, and show that its H-spectral radius and its Z-spectral radius m are smaller than or equal to M (a)nm−1 and M (a)n 2 , respectively, here M (a) is a constant depending on a. Moreover, both infinite and finite dimensional generalized Hilbert tensors are positive definite for a ≥ 1. For an m-order infinite dimensional generalized Hilbert tensor H∞ with a > 0, we prove that H∞ defines a bounded and positively (m − 1)-homogeneous operator from l1 into lp (1 < p < ∞). Two upper bounds on the norms of corresponding positively homogeneous operators are obtained. © 2017 Elsevier Inc. All rights reserved.
✩ This work was supported by the National Natural Science Foundation of China (Grant Nos. 11571095, 11601134). * Corresponding author. E-mail addresses: [email protected] (W. Mei), [email protected] (Y. Song).
http://dx.doi.org/10.1016/j.laa.2017.05.052 0024-3795/© 2017 Elsevier Inc. All rights reserved.
W. Mei, Y. Song / Linear Algebra and its Applications 532 (2017) 8–24
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1. Introduction Let R denote the set of all real numbers, and let Z be the set of all integers. We write Z− for the set of all non-positive integers and Z−− for the set of all negative integers, i.e., Z− = {k ∈ Z : k ≤ 0} and Z−− = {k ∈ Z : k < 0}. An m-order n-dimensional tensor (hypermatrix) A = (ai1 ···im ) is a multi-array of real entries ai1 ···im ∈ R, where ij ∈ In = {1, 2, · · · , n} for j ∈ Im = {1, 2, · · · , m}. A is called a symmetric tensor if the entries ai1 ···im are invariant under any permutation of their indices. Qi [16,17] introduced the concepts of eigenvalues of the higher order symmetric tensors, and showed the existence of the eigenvalues and some applications. The concepts of real eigenvalue was introduced by Lim [14] independently using a variational approach. Subsequently, many mathematicians studied the spectral properties of various structured tensors under different conditions. The spectral properties of nonnegative matrices had been generalized to nonnegative tensors under various conditions by Chang et al. [1,2], He and Huang [8], He [9], He et al. [10], Li et al. [13], Qi [18,20], Song and Qi [22,23], Wang et al. [26], Yang and Yang [28,29] and references therein. Study on infinite dimensional higher order tensors has just begun, and fewer work can be found on this topic. Song and Qi [21] introduced the concept of infinite dimensional Hilbert tensor and showed that such a Hilbert tensor defines a bounded, continuous and positively (m − 1)-homogeneous operator from l1 into lp (1 < p < ∞) and the norms of corresponding positively homogeneous operators are not larger than √π6 . They also proved that the spectral radius and E-spectral radius of finite dimensional Hilbert tensor m are smaller than or equal to nm−1 sin nπ and n 2 sin πn , respectively. Clearly, an m-order 1 n-dimensional Hilbert tensor is a Hankel tensor with v = (1, 12 , 13 , · · · , nm ), introduced by Qi [19]. Also see Chen and Qi [3], Xu [27] for more details of Hankel tensors. A Hilbert tensor (hypermatrix) is a natural extension of Hilbert matrix introduced by Hilbert [7]. For more details of Hilbert matrix, see Frazer [5] and Taussky [25] for n-dimensional Hilbert matrix, Choi [4] and Ingham [11] for an infinite dimensional Hilbert matrix, Magnus [15] and Kato [12] for the spectral properties of infinite dimensional Hilbert matrix. In this paper, we study more general Hilbert tensor, which is referred to as “generalized Hilbert tensor”. The entries of an m-order infinite dimensional generalized Hilbert tensor H∞ = (Hi1 i2 ···im ) are defined by Hi1 i2 ···im =
1 , a ∈ R \ Z− , i1 , i2 , · · · , im ∈ Z++ = −Z−− . i1 + i2 + · · · + im − m + a (1.1)
10
W. Mei, Y. Song / Linear Algebra and its Applications 532 (2017) 8–24
Clearly, both Hn and H∞ are symmetric and the entries of the generalized Hilbert tensor with a ≥ 1 can be written as the following integral 1 Hi1 i2 ···im =
ti1 +i2 +···im −m+a−1 dt.
(1.2)
0
For a vector x = (x1 , x2 , · · · , xn ) ∈ Rn , Hn xm−1 is a vector with its ith component defined by (Hn xm−1 )i =
n i2 ,··· ,im
xi2 · · · xim , a ∈ R \ Z− , i = 1, 2, · · · , n. i + i + · · · + i − m + a 2 m =1
(1.3)
Accordingly, Hn xm is given by n
Hn xm = x (Hn xm−1 ) =
i1 ,i2 ,··· ,im
xi1 xi2 · · · xim , a ∈ R \ Z− . i + i + · · · + i − m + a 1 2 m =1
(1.4)
Let l1 is a linear space consisting of all real number sequences x = (xi )∞ i=1 satisfying ∞
|xi | < ∞.
i=1
For a real vector x = (x1 , x2 , · · · , xn , xn+1 , · · · ) ∈ l1 , H∞ xm−1 is an infinite dimensional vector with its ith component defined by (H∞ xm−1 )i =
∞ i2 ,··· ,im
xi2 · · · xim , a ∈ R \ Z− ; i ∈ {1, 2, · · · , n, · · · }. i + i + · · · + i − m + a 2 m =1 (1.5)
Accordingly, H∞ xm is given by H∞ xm =
∞ i1 ,i2 ,··· ,im
xi1 xi2 · · · xim , a ∈ R \ Z− . i + i + · · · + i − m + a 1 2 m =1
(1.6)
Now we show that both H∞ xm and H∞ xm−1 are well-defined for all x ∈ l1 . Proposition 1.1. Let H∞ be an m-order infinite dimensional generalized Hilbert tensor. Then for all x ∈ l1 , (i) H∞ xm is absolutely convergent, i.e., |H∞ xm | < ∞; (ii) H∞ xm−1 is well-defined, i.e., each of its components absolutely convergent.
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Proof. Let [a] denote the largest integer not exceeding a. Then for all a ∈ R \ Z− , it is obvious that for a > 0, min |i1 + i2 + · · · + im − m + a| = a.
i1 ,··· ,im
For a < 0, there exist some positive integers i1 , i2 , · · · , im and i1 , i2 , · · · , im such that i1 + i2 + · · · + im − m = −[a] and i1 + i2 + · · · + im − m = −[a] − 1, and hence, min |i1 + i2 + · · · + im − m + a| = min{a − [a], 1 − (a − [a])}.
i1 ,··· ,im
In conclusion, we have for a ∈ R \ Z− , 1 ≤ N (a) = |i1 + i2 + · · · + im − m + a|
1 a,
a > 0;
1 min{a−[a],1+[a]−a} ,
a < 0.
Then for x = (x1 , x2 , · · · , xn , xn+1 , · · · ) ∈ l1 , we obtain ∞ xi1 xi2 · · · xim |H∞ xm | = i1 ,i2 ,··· ,im =1 i1 + i2 + · · · + im − m + a ∞
≤
i1 ,i2 ,··· ,im
≤ N (a)
|xi1 xi2 · · · xim | |i + i 2 + · · · + im − m + a| =1 1 ∞
|xi1 ||xi2 | · · · |xim |
i1 ,i2 ,··· ,im =1
= N (a)
∞
m
|xi |
<∞
i=1
∞ since i=1 |xi | < ∞, and hence, H∞ xm is absolutely convergent. Similarly, for each positive integer i, we also have ∞ xi2 · · · xim |(H∞ xm−1 )i | = i2 ,··· ,im =1 i + i2 + · · · + im − m + a ∞ m−1 ≤N (a) |xk | < ∞. k=1
So H∞ xm−1 is well-defined. 2
W. Mei, Y. Song / Linear Algebra and its Applications 532 (2017) 8–24
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In section 2, we present some concepts and basic facts which are used later. In section 3, we first show that both infinite and finite dimensional generalized Hilbert tensors are positive definite for a ≥ 1. Let F∞ x = (H∞ x
1 ] m−1 [ m−1
)
and T∞ x =
H∞ xm−1 xl2−m 1
x = θ
θ
x = θ,
(1.7)
where θ = (0, 0, · · · , 0, · · · ) . Then we prove that both F∞ and T∞ are bounded, continuous and positively homogeneous operators on l1 . In particular, their upper bounds are obtained as follows: F∞ =
sup F∞ xl2(m−1) ≤ K(a) and T∞ =
xl1 =1
sup T∞ xl2 ≤ C(a),
(1.8)
xl1 =1
where ⎧ ⎨ 2(m−1) 12 + a K(a) = ⎩ 2(m−1) π2 6 ,
π2 6 ,
⎧ ⎨
0 < a < 1; and C(a) = a ≥ 1;
1 a2
+
π2 6 ,
⎩ √π , 6
0 < a < 1; a ≥ 1.
For a finite dimensional generalized Hilbert tensor Hn , we show that its H-spectral m radius and its Z-spectral radius are smaller than or equal to M (a)nm−1 and M (a)n 2 , respectively, where ⎧ 1 ⎪ ⎪ ⎨a, M (a) =
a > 0;
1 min{a−[a],1+[a]−a} , ⎪ ⎪ ⎩ 1 −m(n−1)−a ,
−m(n − 1) < a < 0; a < −m(n − 1).
In particular, for a > 0, we obtain an upper bound of its spectral radius ρ(Hn ) and its E-spectral radius ρE (Hn ), respectively, i.e., m
ρ(Hn ) ≤
nm−1 n2 and ρE (Hn ) ≤ . a a
(1.9)
2. Preliminaries and basic facts Let X and Y be two real Banach spaces with the norm · X and · Y , respectively, and let T : X → Y be an operator. Then T is said to be (i) t-homogeneous if T (λx) = λt T (x) for all λ ∈ R and all x ∈ X; (ii) positively homogeneous if T (tx) = tT (x) for all t > 0 and all x ∈ X; (iii) bounded if there exists a real number M > 0 such that T xY ≤ M xX for all x ∈ X.
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For a bounded, continuous and positively homogeneous operator T : X → Y , the norm of T may be defined as follows (see Fucik et al. [6], Song and Qi [23] for more details): T = sup{T xY : xX = 1}.
(2.1)
For 0 < p < ∞, lp is a space consisting of all real number sequences x = (xi )∞ i=1 ∞ satisfying |xii |p < ∞. For p ≥ 1, it is well known that i=1
xlp =
∞
p1 |xi |
p
i=1
is the norm defined on the space lp . For p ≥ 1, it is well known that xp =
n
p1 |xi |
p
i=1
is the norm defined on Rn . Moreover, the following relationship between the two different norms is obvious, xq ≤ xp ≤ n p − q xq for q > p. 1
1
(2.2)
Let A be an m-order n-dimensional symmetric tensor. Then a number λ is called an eigenvalue of A if there exists a nonzero vector x such that Axm−1 = λx[m−1] ,
(2.3)
where x[m−1] = (xm−1 , xm−1 , · · · xm−1 ) , and call x an eigenvector of A associated n 1 2 with λ. We call such an eigenvalue H-eigenvalue if it is real and has a real eigenvector x, and call such a real eigenvector x H-eigenvector. These concepts were first introduced by Qi [16] for the higher order symmetric tensors. Lim [14] independently introduced the notion of eigenvalue for higher order tensors but restricted x to be a real vector and λ to be a real number. Qi [16] introduced another concept of tensor eigenvalue. A number μ is said to be an E-eigenvalue of A if there exists a nonzero vector x such that Axm−1 = μx(xT x)
m−2 2
,
(2.4)
and such a nonzero vector x is called an E-eigenvector of A associated with μ. It is clear that if x is real, then μ is also real. In this case, μ and x are called a Z-eigenvalue of A and a Z-eigenvector of A, respectively. Qi [16,17] extended some nice properties of symmetric matrices to higher order symmetric tensors. The Perron–Frobenius theorem of nonnegative matrices had been generalized to higher order nonnegative tensors under
W. Mei, Y. Song / Linear Algebra and its Applications 532 (2017) 8–24
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various conditions by Chang, Pearson and Zhang [1,2], Qi [18], Song and Qi [22,23], Yang and Yang [28,29] and references therein. 3. Generalized Hilbert tensors In this section, we mainly discuss the properties of infinite and finite dimensional generalized Hilbert tensors. It is easy to see that both finite and infinite dimensional generalized Hilbert tensor Hn and H∞ are symmetric, but not always positive semidefinite since a ∈ R \ Z− . However, for all a > 0, the entries of finite and infinite dimensional generalized Hilbert tensors are positive, then it is obvious that Hn and H∞ are strictly copositive, i.e., Hn xm > 0 for all nonnegative nonzero vector x ∈ Rn , H∞ xm > 0 for all nonnegative nonzero vector x ∈ l1 . The concept of strictly copositive tensors was introduced by Qi [18]. Also see Song and Qi [24] for more details. Now, we give the positive definitiveness of such two tensors when a ≥ 1. 3.1. Positive definiteness of generalized Hilbert tensors Theorem 3.1. Let m be even and a ≥ 1. Then m-order generalized Hilbert tensors H∞ and Hn are both positive definite, i.e., H∞ xm > 0 for all nonzero vector x ∈ l1 , Hn xm > 0 for all nonzero vector x ∈ Rn . Proof. For each x = (x1 , x2 , · · · , xn , xn+1 , · · · ) ∈ l1 and t ∈ [0, 1], it is obvious that |ti−1+
a−1 m
xi | = |t|i−1+
which implies that the infinite series
∞
ti−1+
a−1 m
a−1 m
|xi | ≤ |xi |,
xi uniformly converges with respect to
i=1
t ∈ [0, 1]. Then it follows from the integral form of Hi1 i2 ···im that H∞ x
m
=
∞
1
ti1 +i2 +···im −m+a−1 xi1 xi2 · · · xim dt
i1 ,i2 ,··· ,im =1 0
=
∞
1
i1 ,i2 ,··· ,im =1 0
(ti1 −1+
a−1 m
xi1 )(ti2 −1+
a−1 m
xi2 ) · · · (tim −1+
a−1 m
xim )dt
W. Mei, Y. Song / Linear Algebra and its Applications 532 (2017) 8–24
1 ∞
=
t
xi1
i1 =1
0
1 ∞
=
i1 −1+ a−1 m
∞
t
i2 −1+ a−1 m
xi2
···
i2 =1
∞
15
t
im −1+ a−1 m
xim
dt
im =1
m t
i−1+ a−1 m
dt ≥ 0 (m is even).
xi
i=1
0
Now we show H∞ xm > 0 for all nonzero vector x ∈ l1 . Suppose not, then there exists a nonzero vector x ∈ l1 such that H∞ xm = 0, i.e., 1 ∞
∞
t
xi
dt = 0.
i=1
0
Since m is even,
m i−1+ a−1 m
m ti−1+
a−1 m
≥ 0, and hence, by the continuity, we have
xi
i=1
t
a−1 m
x1 +
∞
t
i−1
xi
∞
=
i=2
ti−1+
a−1 m
xi = 0 for all t ∈ [0, 1].
i=1
So for all t ∈ (0, 1], we have
∞
x1 +
t
i−1
xi
= 0.
i=2
Since the polynomial x1 +
∞
ti−1 xi
is continuous at t = 0, we see that
i=2
x1 +
∞
t
i−1
xi
= 0 for all t ∈ [0, 1].
i=2
Let t = 0. Then x1 = 0, and so, t
a−1 m +1
x2 +
∞
ti−2 xi
=
i=3
∞
ti−1+
a−1 m
xi = 0 for all t ∈ [0, 1].
i=2
So, for all t ∈ (0, 1], we have
x2 +
∞ i=3
ti−2 xi = 0.
W. Mei, Y. Song / Linear Algebra and its Applications 532 (2017) 8–24
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Again by the continuity, we see that x2 +
∞
ti−2 xi = 0 for all t ∈ [0, 1].
i=3
Take t = 0, then we have x2 = 0. By the mathematical induction, suppose x1 = x2 = · · · = xk = 0. Then we only need to show xk+1 = 0. In fact,
t
a−1 m +k
xk+1 +
∞
t
i−(k+1)
xi
=
i=k+2
∞
ti−1+
a−1 m
xi = 0 for all t ∈ [0, 1].
i=k+1
So, for all t ∈ (0, 1], we have xk+1 +
∞
ti−(k+1) xi
= 0.
i=k+2
Again by the continuity, we see that xk+1 +
∞
t
i−(k+1)
xi
= 0 for all t ∈ [0, 1].
i=k+2
Take t = 0, then we have xk+1 = 0. So we may conclude that xi = 0 for all i = 1, 2, · · · , n, n + 1, · · · , and hence, x = θ = (0, 0, · · · , 0, 0, · · · ) , which is a contradiction. The desired conclusion follows. Similarly, we can obtain that Hn is positive definite. 2 Remark 3.1. We show the positive definiteness of infinite and finite dimensional generalized Hilbert tensors with a ≥ 1. Then it is unknown whether or not Hn and H∞ have the positive definiteness for 0 < a < 1 or a < 0 with a ∈ R \ Z− . 3.2. Infinite dimensional generalized Hilbert tensors Let 1
F∞ x = (H∞ xm−1 )[ m−1 ] (m is even)
(3.1)
and T∞ x =
x2−m H∞ xm−1 l1
x = θ
θ
x = θ,
(3.2)
W. Mei, Y. Song / Linear Algebra and its Applications 532 (2017) 8–24 1
1
1
17
1
where x[ m−1 ] = (x1m−1 , x2m−1 , · · · xnm−1 , · · · ) and θ = (0, 0, · · · , 0, · · · ) . It is easy to see that both operators F∞ and T∞ are continuous and positively homogeneous. Therefore our main interests are to study their boundedness. Theorem 3.2. Let F∞ and T∞ be defined by Eqs. (3.1) and (3.2), respectively. Assume that a > 0. (i) If x ∈ l1 , then F∞ x ∈ lp for m − 1 < p < ∞. Moreover, F∞ is a bounded, continuous and positively homogeneous operator from l1 into lp (m − 1 < p < ∞). In particular, F∞ =
sup F∞ xl2(m−1) ≤ K(a),
xl1 =1
where ⎧ ⎨ 2(m−1) 12 + a K(a) = ⎩ 2(m−1) π2 6 ,
π2 6 ,
0 < a < 1; a ≥ 1.
(ii) If x ∈ l1 , then T∞ x ∈ lp for 1 < p < ∞. Moreover, T∞ is a bounded, continuous and positively homogeneous operator from l1 into lp (1 < p < ∞). In particular, T∞ =
sup T∞ xl2 ≤ C(a),
xl1 =1
where ⎧ ⎨ C(a) =
1 a2
⎩ √π , 6
+
π2 6 ,
0 < a < 1; a ≥ 1.
Proof. For x ∈ l1 ,
n x · · · x i2 im m−1 |(H∞ x )i | = lim n→∞ i + i + · · · + i − m + a 2 m i2 ,··· ,im =1 ≤ lim
n→∞
≤ lim
n→∞
n i2 ,··· ,im
|xi2 · · · xim | |i + i + · · · + im − m + a| 2 =1
n
|xi2 ||xi3 | · · · |xim | |i + 1 + · · · + 1 −m + a| i2 ,··· ,im =1 m−1
=
1 lim i − 1 + a n→∞ i
n
2 ,···
,im =1
|xi2 ||xi3 | · · · |xim |
W. Mei, Y. Song / Linear Algebra and its Applications 532 (2017) 8–24
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n m−1 1 = |xk | lim i − 1 + a n→∞ k=1
1 = i−1+a =
∞
m−1 |xk |
k=1
1 . xm−1 l1 i−1+a
Since a > 0, then we have for all positive integer i > 1 1 1 ≤ . (i − 1 + a)s (i − 1)s So the series
∞
1 (i−1+a)s
i=1
converges whenever s > 1.
(i) For m − 1 < p < ∞, we obtain ∞
|(F∞ x)i |p =
i=1
∞ 1 | H∞ xm−1 im−1 |p i=1
∞ p = | H∞ xm−1 i | m−1 i=1
≤
∞ i=1
= xpl1 since s =
p m−1
1 xm−1 l1 (i − 1 + a) ∞
1
i=1
(i − 1 + a) m−1
p
p m−1
<∞
> 1. Thus F∞ x ∈ lp for all x ∈ l1 . Furthermore, we have F∞ xlp =
∞
p1 |(F∞ x)i |p
≤ M xl1 ,
(3.3)
i=1
where M =
∞
p1
1
i=1
p (i−1+a) m−1
. Therefore F∞ is a bounded, continuous and positively
homogeneous operator from l1 into lp (m − 1 < p < ∞). Take p = 2(m − 1), then for a ≥ 1, we have a − 1 ≥ 0, and hence, M=
∞ i=1
1 (i − 1 + a)2
p1
∞ 1 1 p ≤ = i2 i=1
2(m−1)
π2 . 6
W. Mei, Y. Song / Linear Algebra and its Applications 532 (2017) 8–24
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If 0 < a < 1, then M=
∞ i=1
1 (i − 1 + a)2
p1 ≤
∞
1 1 + 2 (1 − 1 + a) (i − 1)2 i=2
p1
2(m−1)
=
π2 1 + . 2 a 6
So we have F∞ =
sup F∞ xl2(m−1) ≤ M ≤ K(a).
xl1 =1
(ii) For 1 < p < ∞, we have ∞
|(T∞ x)i |p =
i=1
∞
|(x2−m H∞ xm−1 )i |p l1
i=1 (2−m)p
= xl1
∞
|(H∞ xm−1 )i |p
i=1
∞ (2−m)p
≤ xl1
i=1
= xpl1
∞ i=1
1 xm−1 l1 (i − 1 + a)
p
1 <∞ (i − 1 + a)p
since s = p > 1. Thus, T∞ x ∈ lp for all x ∈ l1 . Furthermore, we have ∞ 1 T∞ xlp = ( |(T∞ x)i |p ) p ≤ Cxl1 ,
(3.4)
i=1
where C =
∞ i=1
1 (i−1+a)p
p1 > 0. So T∞ is a bounded, continuous and positively homo-
geneous operator from l1 into lp (1 < p < ∞). Similarly, take p = 2, then for a ≥ 1, we have C=
∞ i=1
1 (i − 1 + a)2
12
≤
∞ 1 i2 i=1
12
≤
π2 . 6
If 0 < a < 1, then C=
∞ i=1
1 (i − 1 + a)2
12
≤
∞
1 1 + 2 (1 − 1 + a) (i − 1)2 i=2
12
=
1 π2 + . 2 a 6
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20
So we obtain T∞ =
sup T∞ xl2 ≤ C ≤ C(a).
xl1 =1
The desired results follow. 2 From the proof of Theorem 3.2, it is easy to obtain the following properties of operator defined by infinite dimensional generalized Hilbert tensor H∞ . Theorem 3.3. Let f (x) = H∞ xm−1 and a > 0. Then f is a bounded, continuous and positively (m − 1)-homogeneous operator from l1 into lp (1 < p < ∞). Remark 3.2. (i) In Theorem 3.3, H∞ defines a bounded operator for a > 0. Then it is unknown whether or not H∞ is bounded for a < 0 with a ∈ R \ Z− . (ii) For an m-order infinite dimensional generalized Hilbert tensor, two upper bounds on the norms of corresponding positively homogeneous operators are showed for a > 0, then for a < 0 with a ∈ R \ Z− , it is unknown whether have similar conclusions or not. (iii) Are the upper bounds the best in Theorem 3.2? 3.3. Finite dimensional generalized Hilbert tensors Theorem 3.4. Let Hn be an m-order n-dimensional generalized Hilbert tensor. Assume that a ∈ R \ Z− and ⎧ 1 ⎪ ⎪ ⎨a, M (a) =
1 min{a−[a],1+[a]−a} , ⎪ ⎪ ⎩ 1 −m(n−1)−a ,
a > 0; −m(n − 1) < a < 0; a < −m(n − 1),
where [a] is the largest integer not exceeding a. Then (i) |λ| ≤ M (a)nm−1 for all H-eigenvalues λ of Hn if m is even; m (ii) |μ| ≤ M (a)n 2 for all Z-eigenvalues μ of Hn . Proof. Using the proof technique of Proposition 1.1, we have |i1 + i2 + · · · + im − m + a| ≥ a for a > 0, |i1 + i2 + · · · + im − m + a| ≥ min{a − [a], 1 − (a − [a])} for − m(n − 1) < a < 0, |i1 + i2 + · · · + im − m + a| ≥ −m(n − 1) − a for a < −m(n − 1).
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In conclusion, 1 ≤ M (a). |i1 + i2 + · · · + im − m + a| Then for each nonzero vector x ∈ Rn , we have n x x · · · x i i i 1 2 m m |Hn x | = i + i + · · · + i − m + a 2 m i1 ,i2 ,··· ,im =1 1 n
≤
i1 ,i2 ,··· ,im
|xi1 xi2 · · · xim | |i + i + · · · + im − m + a| 1 2 =1 n
≤ M (a)
|xi1 ||xi2 | · · · , |xim |
i1 ,i2 ,··· ,im =1 n |xi |)m = M (a)( i=1
= M (a)xm 1 . 1
(i) By the fact x1 ≤ n1− m xm , we obtain 1
1− m |Hn xm | ≤ M (a)xm xm )m 1 ≤ M (a)(n
= M (a)nm−1 xm m, and hence, we have for all nonzero vector x ∈ Rn m |Hn xm | x Hn = ≤ M (a)nm−1 . xm xm m
(3.5)
Let λ be an H-eigenvalues of Hn . Then there exists a nonzero vector y ∈ Rn such that Hn y m−1 = λy [m−1] , and so, Hn y m = y (Hn y m−1 ) = λy y [m−1] = λ
n
yim = λym m
i=1
since m is even. Thus, it follows from (3.5) that m y |Hn y m | ≤ M (a)nm−1 . |λ| = = Hn ym y m m Then (i) follows since λ is a H-eigenvalue.
W. Mei, Y. Song / Linear Algebra and its Applications 532 (2017) 8–24
22
(ii) Similarly, we have by the fact x1 ≤
√
nx2 m
m 2 |Hn xm | ≤ M (a)xm 1 ≤ M (a)n x2 ,
and hence, we have for all nonzero vector x ∈ Rn m |Hn xm | m x Hn = ≤ M (a)n 2 . x2 xm 2
(3.6)
Let μ be a Z-eigenvalues of Hn . Then there exists a nonzero vector z ∈ Rn such that Hn z m−1 = μz(z z)
m−2 2
,
and so,
Hn z
m
= z (Hn z
m−1
) = μz z(z z)
m−2 2
=μ
n
m2 zi2
= μzm 2 .
i=1
Thus, by (3.6), we obtain m |Hn z m | z ≤ M (a)n m2 . |μ| = = Hn m z2 z2 Then the desired conclusion follows since μ is a Z-eigenvalue. 2 Let a > 0. Then the generalized Hilbert tensor Hn is positive (all entries are positive) and symmetric, and hence, its spectral radius ρ(Hn ) and its E-spectral radius ρE (Hn ) are its H-eigenvalue and its Z-eigenvalue, respectively. For more details, see Chang, Pearson and Zhang [1,2], Qi [18], Song and Qi [22,23] and Yang and Yang [28,29]. So the following collusions are easy to obtain by Theorem 3.4. Corollary 3.5. Let Hn be an m-order n-dimensional generalized Hilbert tensor. Assume that a > 0. Then (i) |λ| ≤ ρ(Hn ) ≤ (ii) |μ| ≤ ρE (Hn )
nm−1 am ≤ na2
for all eigenvalues λ of Hn if m is even; for all E-eigenvalues μ of Hn .
Remark 3.3. We prove the upper bounds of spectral radii in the above results. These upper bounds may not be the best, then the following problems deserve us further research: What is the best upper bounds for the spectral radii of generalized Hilbert tensors? How to compute such an upper bound?
W. Mei, Y. Song / Linear Algebra and its Applications 532 (2017) 8–24
23
Acknowledgements The authors would like to thank the anonymous referees/editors for their valuable suggestions which helped us to improve this manuscript. References [1] K.C. Chang, K. Pearson, T. Zhang, Some variational principles for Z-eigenvalues of nonnegative tensors, Linear Algebra Appl. 438 (2013) 4166–4182. [2] K.C. Chang, K. Pearson, T. Zhang, Perron–Frobenius theorem for nonnegative tensors, Commun. Math. Sci. 6 (2008) 507–520. [3] H. Chen, L. Qi, Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors, J. Ind. Manag. Optim. 11 (4) (2015) 1263–1274. [4] Man-Duen Choi, Tricks for treats with the Hilbert matrix, Amer. Math. Monthly 90 (1983) 301–312. [5] H. Frazer, Note on Hilbert’s inequality, J. Lond. Math. Soc. s1-21 (1) (1946) 7–9. [6] S. Fucik, J. Necas, J. Soucek, V. Soucek, Spectral Analysis of Nonlinear Operators, Lecture Notes in Mathematics, vol. 346, Springer-Verlag, Berlin, Heidelberg, New York, 1973. [7] D. Hilbert, Ein Beitrag zur Theorie des Legendre’schen Polynoms, Acta Math. (Springer Netherlands) 18 (1894) 155–159. [8] J. He, T. Huang, Upper bound for the largest Z-eigenvalue of positive tensors, Appl. Math. Lett. 38 (2014) 110–114. [9] J. He, Bounds for the largest eigenvalue of nonnegative tensors, J. Comput. Anal. Appl. 20 (7) (2016) 1290–1301. [10] J. He, Y.M. Liu, H. Ke, J.K. Tian, X. Li, Bounds for the Z-spectral radius of nonnegative tensors, SpringerPlus 5 (1) (2016) 1727. [11] A.E. Ingham, A note on Hilbert’s inequality, J. Lond. Math. Soc. s1-11 (3) (1936) 237–240. [12] T. Kato, On the Hilbert matrix, Proc. Amer. Math. Soc. 8 (1) (1957) 73–81. [13] W. Li, D. Liu, S.W. Vong, Z-eigenpair bounds for an irreducible nonnegative tensor, Linear Algebra Appl. 483 (2015) 182–199. [14] L.H. Lim, Singular values and eigenvalues of tensors: a variational approach, in: Proc. 1st IEEE International Workshop on Computational Advances of Multi-Tensor Adaptive Processing, Dec. 13–15, 2005, pp. 129–132. [15] W. Magnus, On the spectrum of Hilbert’s matrix, Amer. J. Math. 72 (4) (1950) 699–704. [16] L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput. 40 (2005) 1302–1324. [17] L. Qi, Rank and eigenvalues of a supersymmetric tensor, the multivariate homogeneous polynomial and the algebraic hypersurface it defines, J. Symbolic Comput. 41 (2006) 1309–1327. [18] L. Qi, Symmetric nonnegative tensors and copositive tensors, Linear Algebra Appl. 439 (2013) 228–238. [19] L. Qi, Hankel tensors: associated Hankel matrices and Vandermonde decomposition, Commun. Math. Sci. 13 (1) (2015) 113–125. [20] L. Qi, Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, Society for Industrial & Applied Mathematics, April 19, 2017. [21] Y. Song, L. Qi, Infinite and finite dimensional Hilbert tensors, Linear Algebra Appl. 451 (2014) 1–14. [22] Y. Song, L. Qi, Positive eigenvalue-eigenvector of nonlinear positive mappings, Front. Math. China 9 (1) (2014) 181–199. [23] Y. Song, L. Qi, Spectral properties of positively homogeneous operators induced by higher order tensors, SIAM J. Matrix Anal. Appl. 34 (4) (2013) 1581–1595. [24] Y. Song, L. Qi, Necessary and sufficient conditions for copositive tensors, Linear Multilinear Algebra 63 (1) (2015) 120–131. [25] O. Taussky, A remark concerning the characteristic roots of the finite segments of the Hilbert matrix, Quart. J. Math. os-20 (1) (1949) 80–83. [26] Y. Wang, K. Zhang, H. Sun, Criteria for strong H-tensors, Front. Math. China 11 (3) (2016) 577–592.
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[27] C. Xu, Hankel tensors, Vandermonde tensors and their positivities, Linear Algebra Appl. 491 (2016) 56–72. [28] Q. Yang, Y. Yang, Further results for Perron–Frobenius theorem for nonnegative tensors II, SIAM J. Matrix Anal. Appl. 32 (4) (2011) 1236–1250. [29] Y. Yang, Q. Yang, Further results for Perron–Frobenius theorem for nonnegative tensors, SIAM J. Matrix Anal. Appl. 31 (5) (2010) 2517–2530.
Contents lists available at ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
Infinite and finite dimensional generalized Hilbert tensors ✩ Wei Mei a , Yisheng Song b,∗ a
School of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan, 453007, PR China b School of Mathematics and Information Science and Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control, Henan Normal University, Xinxiang, Henan, 453007, PR China
a r t i c l e
i n f o
Article history: Received 16 November 2016 Accepted 11 May 2017 Available online 23 June 2017 Submitted by R. Brualdi MSC: 47H15 47H12 34B10 47A52 47J10 47H09 15A48 47H07 Keywords: Generalized Hilbert tensor Spectral radius Norm Upper bound
a b s t r a c t In this paper, we introduce the concept of an m-order n-dimensional generalized Hilbert tensor Hn = (Hi1 i2 ···im ), Hi1 i2 ···im =
1 , i1 + i2 + · · · i m − m + a
a ∈ R \ Z− ; i1 , i2 , · · · , im = 1, 2, · · · , n, and show that its H-spectral radius and its Z-spectral radius m are smaller than or equal to M (a)nm−1 and M (a)n 2 , respectively, here M (a) is a constant depending on a. Moreover, both infinite and finite dimensional generalized Hilbert tensors are positive definite for a ≥ 1. For an m-order infinite dimensional generalized Hilbert tensor H∞ with a > 0, we prove that H∞ defines a bounded and positively (m − 1)-homogeneous operator from l1 into lp (1 < p < ∞). Two upper bounds on the norms of corresponding positively homogeneous operators are obtained. © 2017 Elsevier Inc. All rights reserved.
✩ This work was supported by the National Natural Science Foundation of China (Grant Nos. 11571095, 11601134). * Corresponding author. E-mail addresses: [email protected] (W. Mei), [email protected] (Y. Song).
http://dx.doi.org/10.1016/j.laa.2017.05.052 0024-3795/© 2017 Elsevier Inc. All rights reserved.
W. Mei, Y. Song / Linear Algebra and its Applications 532 (2017) 8–24
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1. Introduction Let R denote the set of all real numbers, and let Z be the set of all integers. We write Z− for the set of all non-positive integers and Z−− for the set of all negative integers, i.e., Z− = {k ∈ Z : k ≤ 0} and Z−− = {k ∈ Z : k < 0}. An m-order n-dimensional tensor (hypermatrix) A = (ai1 ···im ) is a multi-array of real entries ai1 ···im ∈ R, where ij ∈ In = {1, 2, · · · , n} for j ∈ Im = {1, 2, · · · , m}. A is called a symmetric tensor if the entries ai1 ···im are invariant under any permutation of their indices. Qi [16,17] introduced the concepts of eigenvalues of the higher order symmetric tensors, and showed the existence of the eigenvalues and some applications. The concepts of real eigenvalue was introduced by Lim [14] independently using a variational approach. Subsequently, many mathematicians studied the spectral properties of various structured tensors under different conditions. The spectral properties of nonnegative matrices had been generalized to nonnegative tensors under various conditions by Chang et al. [1,2], He and Huang [8], He [9], He et al. [10], Li et al. [13], Qi [18,20], Song and Qi [22,23], Wang et al. [26], Yang and Yang [28,29] and references therein. Study on infinite dimensional higher order tensors has just begun, and fewer work can be found on this topic. Song and Qi [21] introduced the concept of infinite dimensional Hilbert tensor and showed that such a Hilbert tensor defines a bounded, continuous and positively (m − 1)-homogeneous operator from l1 into lp (1 < p < ∞) and the norms of corresponding positively homogeneous operators are not larger than √π6 . They also proved that the spectral radius and E-spectral radius of finite dimensional Hilbert tensor m are smaller than or equal to nm−1 sin nπ and n 2 sin πn , respectively. Clearly, an m-order 1 n-dimensional Hilbert tensor is a Hankel tensor with v = (1, 12 , 13 , · · · , nm ), introduced by Qi [19]. Also see Chen and Qi [3], Xu [27] for more details of Hankel tensors. A Hilbert tensor (hypermatrix) is a natural extension of Hilbert matrix introduced by Hilbert [7]. For more details of Hilbert matrix, see Frazer [5] and Taussky [25] for n-dimensional Hilbert matrix, Choi [4] and Ingham [11] for an infinite dimensional Hilbert matrix, Magnus [15] and Kato [12] for the spectral properties of infinite dimensional Hilbert matrix. In this paper, we study more general Hilbert tensor, which is referred to as “generalized Hilbert tensor”. The entries of an m-order infinite dimensional generalized Hilbert tensor H∞ = (Hi1 i2 ···im ) are defined by Hi1 i2 ···im =
1 , a ∈ R \ Z− , i1 , i2 , · · · , im ∈ Z++ = −Z−− . i1 + i2 + · · · + im − m + a (1.1)
10
W. Mei, Y. Song / Linear Algebra and its Applications 532 (2017) 8–24
Clearly, both Hn and H∞ are symmetric and the entries of the generalized Hilbert tensor with a ≥ 1 can be written as the following integral 1 Hi1 i2 ···im =
ti1 +i2 +···im −m+a−1 dt.
(1.2)
0
For a vector x = (x1 , x2 , · · · , xn ) ∈ Rn , Hn xm−1 is a vector with its ith component defined by (Hn xm−1 )i =
n i2 ,··· ,im
xi2 · · · xim , a ∈ R \ Z− , i = 1, 2, · · · , n. i + i + · · · + i − m + a 2 m =1
(1.3)
Accordingly, Hn xm is given by n
Hn xm = x (Hn xm−1 ) =
i1 ,i2 ,··· ,im
xi1 xi2 · · · xim , a ∈ R \ Z− . i + i + · · · + i − m + a 1 2 m =1
(1.4)
Let l1 is a linear space consisting of all real number sequences x = (xi )∞ i=1 satisfying ∞
|xi | < ∞.
i=1
For a real vector x = (x1 , x2 , · · · , xn , xn+1 , · · · ) ∈ l1 , H∞ xm−1 is an infinite dimensional vector with its ith component defined by (H∞ xm−1 )i =
∞ i2 ,··· ,im
xi2 · · · xim , a ∈ R \ Z− ; i ∈ {1, 2, · · · , n, · · · }. i + i + · · · + i − m + a 2 m =1 (1.5)
Accordingly, H∞ xm is given by H∞ xm =
∞ i1 ,i2 ,··· ,im
xi1 xi2 · · · xim , a ∈ R \ Z− . i + i + · · · + i − m + a 1 2 m =1
(1.6)
Now we show that both H∞ xm and H∞ xm−1 are well-defined for all x ∈ l1 . Proposition 1.1. Let H∞ be an m-order infinite dimensional generalized Hilbert tensor. Then for all x ∈ l1 , (i) H∞ xm is absolutely convergent, i.e., |H∞ xm | < ∞; (ii) H∞ xm−1 is well-defined, i.e., each of its components absolutely convergent.
W. Mei, Y. Song / Linear Algebra and its Applications 532 (2017) 8–24
11
Proof. Let [a] denote the largest integer not exceeding a. Then for all a ∈ R \ Z− , it is obvious that for a > 0, min |i1 + i2 + · · · + im − m + a| = a.
i1 ,··· ,im
For a < 0, there exist some positive integers i1 , i2 , · · · , im and i1 , i2 , · · · , im such that i1 + i2 + · · · + im − m = −[a] and i1 + i2 + · · · + im − m = −[a] − 1, and hence, min |i1 + i2 + · · · + im − m + a| = min{a − [a], 1 − (a − [a])}.
i1 ,··· ,im
In conclusion, we have for a ∈ R \ Z− , 1 ≤ N (a) = |i1 + i2 + · · · + im − m + a|
1 a,
a > 0;
1 min{a−[a],1+[a]−a} ,
a < 0.
Then for x = (x1 , x2 , · · · , xn , xn+1 , · · · ) ∈ l1 , we obtain ∞ xi1 xi2 · · · xim |H∞ xm | = i1 ,i2 ,··· ,im =1 i1 + i2 + · · · + im − m + a ∞
≤
i1 ,i2 ,··· ,im
≤ N (a)
|xi1 xi2 · · · xim | |i + i 2 + · · · + im − m + a| =1 1 ∞
|xi1 ||xi2 | · · · |xim |
i1 ,i2 ,··· ,im =1
= N (a)
∞
m
|xi |
<∞
i=1
∞ since i=1 |xi | < ∞, and hence, H∞ xm is absolutely convergent. Similarly, for each positive integer i, we also have ∞ xi2 · · · xim |(H∞ xm−1 )i | = i2 ,··· ,im =1 i + i2 + · · · + im − m + a ∞ m−1 ≤N (a) |xk | < ∞. k=1
So H∞ xm−1 is well-defined. 2
W. Mei, Y. Song / Linear Algebra and its Applications 532 (2017) 8–24
12
In section 2, we present some concepts and basic facts which are used later. In section 3, we first show that both infinite and finite dimensional generalized Hilbert tensors are positive definite for a ≥ 1. Let F∞ x = (H∞ x
1 ] m−1 [ m−1
)
and T∞ x =
H∞ xm−1 xl2−m 1
x = θ
θ
x = θ,
(1.7)
where θ = (0, 0, · · · , 0, · · · ) . Then we prove that both F∞ and T∞ are bounded, continuous and positively homogeneous operators on l1 . In particular, their upper bounds are obtained as follows: F∞ =
sup F∞ xl2(m−1) ≤ K(a) and T∞ =
xl1 =1
sup T∞ xl2 ≤ C(a),
(1.8)
xl1 =1
where ⎧ ⎨ 2(m−1) 12 + a K(a) = ⎩ 2(m−1) π2 6 ,
π2 6 ,
⎧ ⎨
0 < a < 1; and C(a) = a ≥ 1;
1 a2
+
π2 6 ,
⎩ √π , 6
0 < a < 1; a ≥ 1.
For a finite dimensional generalized Hilbert tensor Hn , we show that its H-spectral m radius and its Z-spectral radius are smaller than or equal to M (a)nm−1 and M (a)n 2 , respectively, where ⎧ 1 ⎪ ⎪ ⎨a, M (a) =
a > 0;
1 min{a−[a],1+[a]−a} , ⎪ ⎪ ⎩ 1 −m(n−1)−a ,
−m(n − 1) < a < 0; a < −m(n − 1).
In particular, for a > 0, we obtain an upper bound of its spectral radius ρ(Hn ) and its E-spectral radius ρE (Hn ), respectively, i.e., m
ρ(Hn ) ≤
nm−1 n2 and ρE (Hn ) ≤ . a a
(1.9)
2. Preliminaries and basic facts Let X and Y be two real Banach spaces with the norm · X and · Y , respectively, and let T : X → Y be an operator. Then T is said to be (i) t-homogeneous if T (λx) = λt T (x) for all λ ∈ R and all x ∈ X; (ii) positively homogeneous if T (tx) = tT (x) for all t > 0 and all x ∈ X; (iii) bounded if there exists a real number M > 0 such that T xY ≤ M xX for all x ∈ X.
W. Mei, Y. Song / Linear Algebra and its Applications 532 (2017) 8–24
13
For a bounded, continuous and positively homogeneous operator T : X → Y , the norm of T may be defined as follows (see Fucik et al. [6], Song and Qi [23] for more details): T = sup{T xY : xX = 1}.
(2.1)
For 0 < p < ∞, lp is a space consisting of all real number sequences x = (xi )∞ i=1 ∞ satisfying |xii |p < ∞. For p ≥ 1, it is well known that i=1
xlp =
∞
p1 |xi |
p
i=1
is the norm defined on the space lp . For p ≥ 1, it is well known that xp =
n
p1 |xi |
p
i=1
is the norm defined on Rn . Moreover, the following relationship between the two different norms is obvious, xq ≤ xp ≤ n p − q xq for q > p. 1
1
(2.2)
Let A be an m-order n-dimensional symmetric tensor. Then a number λ is called an eigenvalue of A if there exists a nonzero vector x such that Axm−1 = λx[m−1] ,
(2.3)
where x[m−1] = (xm−1 , xm−1 , · · · xm−1 ) , and call x an eigenvector of A associated n 1 2 with λ. We call such an eigenvalue H-eigenvalue if it is real and has a real eigenvector x, and call such a real eigenvector x H-eigenvector. These concepts were first introduced by Qi [16] for the higher order symmetric tensors. Lim [14] independently introduced the notion of eigenvalue for higher order tensors but restricted x to be a real vector and λ to be a real number. Qi [16] introduced another concept of tensor eigenvalue. A number μ is said to be an E-eigenvalue of A if there exists a nonzero vector x such that Axm−1 = μx(xT x)
m−2 2
,
(2.4)
and such a nonzero vector x is called an E-eigenvector of A associated with μ. It is clear that if x is real, then μ is also real. In this case, μ and x are called a Z-eigenvalue of A and a Z-eigenvector of A, respectively. Qi [16,17] extended some nice properties of symmetric matrices to higher order symmetric tensors. The Perron–Frobenius theorem of nonnegative matrices had been generalized to higher order nonnegative tensors under
W. Mei, Y. Song / Linear Algebra and its Applications 532 (2017) 8–24
14
various conditions by Chang, Pearson and Zhang [1,2], Qi [18], Song and Qi [22,23], Yang and Yang [28,29] and references therein. 3. Generalized Hilbert tensors In this section, we mainly discuss the properties of infinite and finite dimensional generalized Hilbert tensors. It is easy to see that both finite and infinite dimensional generalized Hilbert tensor Hn and H∞ are symmetric, but not always positive semidefinite since a ∈ R \ Z− . However, for all a > 0, the entries of finite and infinite dimensional generalized Hilbert tensors are positive, then it is obvious that Hn and H∞ are strictly copositive, i.e., Hn xm > 0 for all nonnegative nonzero vector x ∈ Rn , H∞ xm > 0 for all nonnegative nonzero vector x ∈ l1 . The concept of strictly copositive tensors was introduced by Qi [18]. Also see Song and Qi [24] for more details. Now, we give the positive definitiveness of such two tensors when a ≥ 1. 3.1. Positive definiteness of generalized Hilbert tensors Theorem 3.1. Let m be even and a ≥ 1. Then m-order generalized Hilbert tensors H∞ and Hn are both positive definite, i.e., H∞ xm > 0 for all nonzero vector x ∈ l1 , Hn xm > 0 for all nonzero vector x ∈ Rn . Proof. For each x = (x1 , x2 , · · · , xn , xn+1 , · · · ) ∈ l1 and t ∈ [0, 1], it is obvious that |ti−1+
a−1 m
xi | = |t|i−1+
which implies that the infinite series
∞
ti−1+
a−1 m
a−1 m
|xi | ≤ |xi |,
xi uniformly converges with respect to
i=1
t ∈ [0, 1]. Then it follows from the integral form of Hi1 i2 ···im that H∞ x
m
=
∞
1
ti1 +i2 +···im −m+a−1 xi1 xi2 · · · xim dt
i1 ,i2 ,··· ,im =1 0
=
∞
1
i1 ,i2 ,··· ,im =1 0
(ti1 −1+
a−1 m
xi1 )(ti2 −1+
a−1 m
xi2 ) · · · (tim −1+
a−1 m
xim )dt
W. Mei, Y. Song / Linear Algebra and its Applications 532 (2017) 8–24
1 ∞
=
t
xi1
i1 =1
0
1 ∞
=
i1 −1+ a−1 m
∞
t
i2 −1+ a−1 m
xi2
···
i2 =1
∞
15
t
im −1+ a−1 m
xim
dt
im =1
m t
i−1+ a−1 m
dt ≥ 0 (m is even).
xi
i=1
0
Now we show H∞ xm > 0 for all nonzero vector x ∈ l1 . Suppose not, then there exists a nonzero vector x ∈ l1 such that H∞ xm = 0, i.e., 1 ∞
∞
t
xi
dt = 0.
i=1
0
Since m is even,
m i−1+ a−1 m
m ti−1+
a−1 m
≥ 0, and hence, by the continuity, we have
xi
i=1
t
a−1 m
x1 +
∞
t
i−1
xi
∞
=
i=2
ti−1+
a−1 m
xi = 0 for all t ∈ [0, 1].
i=1
So for all t ∈ (0, 1], we have
∞
x1 +
t
i−1
xi
= 0.
i=2
Since the polynomial x1 +
∞
ti−1 xi
is continuous at t = 0, we see that
i=2
x1 +
∞
t
i−1
xi
= 0 for all t ∈ [0, 1].
i=2
Let t = 0. Then x1 = 0, and so, t
a−1 m +1
x2 +
∞
ti−2 xi
=
i=3
∞
ti−1+
a−1 m
xi = 0 for all t ∈ [0, 1].
i=2
So, for all t ∈ (0, 1], we have
x2 +
∞ i=3
ti−2 xi = 0.
W. Mei, Y. Song / Linear Algebra and its Applications 532 (2017) 8–24
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Again by the continuity, we see that x2 +
∞
ti−2 xi = 0 for all t ∈ [0, 1].
i=3
Take t = 0, then we have x2 = 0. By the mathematical induction, suppose x1 = x2 = · · · = xk = 0. Then we only need to show xk+1 = 0. In fact,
t
a−1 m +k
xk+1 +
∞
t
i−(k+1)
xi
=
i=k+2
∞
ti−1+
a−1 m
xi = 0 for all t ∈ [0, 1].
i=k+1
So, for all t ∈ (0, 1], we have xk+1 +
∞
ti−(k+1) xi
= 0.
i=k+2
Again by the continuity, we see that xk+1 +
∞
t
i−(k+1)
xi
= 0 for all t ∈ [0, 1].
i=k+2
Take t = 0, then we have xk+1 = 0. So we may conclude that xi = 0 for all i = 1, 2, · · · , n, n + 1, · · · , and hence, x = θ = (0, 0, · · · , 0, 0, · · · ) , which is a contradiction. The desired conclusion follows. Similarly, we can obtain that Hn is positive definite. 2 Remark 3.1. We show the positive definiteness of infinite and finite dimensional generalized Hilbert tensors with a ≥ 1. Then it is unknown whether or not Hn and H∞ have the positive definiteness for 0 < a < 1 or a < 0 with a ∈ R \ Z− . 3.2. Infinite dimensional generalized Hilbert tensors Let 1
F∞ x = (H∞ xm−1 )[ m−1 ] (m is even)
(3.1)
and T∞ x =
x2−m H∞ xm−1 l1
x = θ
θ
x = θ,
(3.2)
W. Mei, Y. Song / Linear Algebra and its Applications 532 (2017) 8–24 1
1
1
17
1
where x[ m−1 ] = (x1m−1 , x2m−1 , · · · xnm−1 , · · · ) and θ = (0, 0, · · · , 0, · · · ) . It is easy to see that both operators F∞ and T∞ are continuous and positively homogeneous. Therefore our main interests are to study their boundedness. Theorem 3.2. Let F∞ and T∞ be defined by Eqs. (3.1) and (3.2), respectively. Assume that a > 0. (i) If x ∈ l1 , then F∞ x ∈ lp for m − 1 < p < ∞. Moreover, F∞ is a bounded, continuous and positively homogeneous operator from l1 into lp (m − 1 < p < ∞). In particular, F∞ =
sup F∞ xl2(m−1) ≤ K(a),
xl1 =1
where ⎧ ⎨ 2(m−1) 12 + a K(a) = ⎩ 2(m−1) π2 6 ,
π2 6 ,
0 < a < 1; a ≥ 1.
(ii) If x ∈ l1 , then T∞ x ∈ lp for 1 < p < ∞. Moreover, T∞ is a bounded, continuous and positively homogeneous operator from l1 into lp (1 < p < ∞). In particular, T∞ =
sup T∞ xl2 ≤ C(a),
xl1 =1
where ⎧ ⎨ C(a) =
1 a2
⎩ √π , 6
+
π2 6 ,
0 < a < 1; a ≥ 1.
Proof. For x ∈ l1 ,
n x · · · x i2 im m−1 |(H∞ x )i | = lim n→∞ i + i + · · · + i − m + a 2 m i2 ,··· ,im =1 ≤ lim
n→∞
≤ lim
n→∞
n i2 ,··· ,im
|xi2 · · · xim | |i + i + · · · + im − m + a| 2 =1
n
|xi2 ||xi3 | · · · |xim | |i + 1 + · · · + 1 −m + a| i2 ,··· ,im =1 m−1
=
1 lim i − 1 + a n→∞ i
n
2 ,···
,im =1
|xi2 ||xi3 | · · · |xim |
W. Mei, Y. Song / Linear Algebra and its Applications 532 (2017) 8–24
18
n m−1 1 = |xk | lim i − 1 + a n→∞ k=1
1 = i−1+a =
∞
m−1 |xk |
k=1
1 . xm−1 l1 i−1+a
Since a > 0, then we have for all positive integer i > 1 1 1 ≤ . (i − 1 + a)s (i − 1)s So the series
∞
1 (i−1+a)s
i=1
converges whenever s > 1.
(i) For m − 1 < p < ∞, we obtain ∞
|(F∞ x)i |p =
i=1
∞ 1 | H∞ xm−1 im−1 |p i=1
∞ p = | H∞ xm−1 i | m−1 i=1
≤
∞ i=1
= xpl1 since s =
p m−1
1 xm−1 l1 (i − 1 + a) ∞
1
i=1
(i − 1 + a) m−1
p
p m−1
<∞
> 1. Thus F∞ x ∈ lp for all x ∈ l1 . Furthermore, we have F∞ xlp =
∞
p1 |(F∞ x)i |p
≤ M xl1 ,
(3.3)
i=1
where M =
∞
p1
1
i=1
p (i−1+a) m−1
. Therefore F∞ is a bounded, continuous and positively
homogeneous operator from l1 into lp (m − 1 < p < ∞). Take p = 2(m − 1), then for a ≥ 1, we have a − 1 ≥ 0, and hence, M=
∞ i=1
1 (i − 1 + a)2
p1
∞ 1 1 p ≤ = i2 i=1
2(m−1)
π2 . 6
W. Mei, Y. Song / Linear Algebra and its Applications 532 (2017) 8–24
19
If 0 < a < 1, then M=
∞ i=1
1 (i − 1 + a)2
p1 ≤
∞
1 1 + 2 (1 − 1 + a) (i − 1)2 i=2
p1
2(m−1)
=
π2 1 + . 2 a 6
So we have F∞ =
sup F∞ xl2(m−1) ≤ M ≤ K(a).
xl1 =1
(ii) For 1 < p < ∞, we have ∞
|(T∞ x)i |p =
i=1
∞
|(x2−m H∞ xm−1 )i |p l1
i=1 (2−m)p
= xl1
∞
|(H∞ xm−1 )i |p
i=1
∞ (2−m)p
≤ xl1
i=1
= xpl1
∞ i=1
1 xm−1 l1 (i − 1 + a)
p
1 <∞ (i − 1 + a)p
since s = p > 1. Thus, T∞ x ∈ lp for all x ∈ l1 . Furthermore, we have ∞ 1 T∞ xlp = ( |(T∞ x)i |p ) p ≤ Cxl1 ,
(3.4)
i=1
where C =
∞ i=1
1 (i−1+a)p
p1 > 0. So T∞ is a bounded, continuous and positively homo-
geneous operator from l1 into lp (1 < p < ∞). Similarly, take p = 2, then for a ≥ 1, we have C=
∞ i=1
1 (i − 1 + a)2
12
≤
∞ 1 i2 i=1
12
≤
π2 . 6
If 0 < a < 1, then C=
∞ i=1
1 (i − 1 + a)2
12
≤
∞
1 1 + 2 (1 − 1 + a) (i − 1)2 i=2
12
=
1 π2 + . 2 a 6
W. Mei, Y. Song / Linear Algebra and its Applications 532 (2017) 8–24
20
So we obtain T∞ =
sup T∞ xl2 ≤ C ≤ C(a).
xl1 =1
The desired results follow. 2 From the proof of Theorem 3.2, it is easy to obtain the following properties of operator defined by infinite dimensional generalized Hilbert tensor H∞ . Theorem 3.3. Let f (x) = H∞ xm−1 and a > 0. Then f is a bounded, continuous and positively (m − 1)-homogeneous operator from l1 into lp (1 < p < ∞). Remark 3.2. (i) In Theorem 3.3, H∞ defines a bounded operator for a > 0. Then it is unknown whether or not H∞ is bounded for a < 0 with a ∈ R \ Z− . (ii) For an m-order infinite dimensional generalized Hilbert tensor, two upper bounds on the norms of corresponding positively homogeneous operators are showed for a > 0, then for a < 0 with a ∈ R \ Z− , it is unknown whether have similar conclusions or not. (iii) Are the upper bounds the best in Theorem 3.2? 3.3. Finite dimensional generalized Hilbert tensors Theorem 3.4. Let Hn be an m-order n-dimensional generalized Hilbert tensor. Assume that a ∈ R \ Z− and ⎧ 1 ⎪ ⎪ ⎨a, M (a) =
1 min{a−[a],1+[a]−a} , ⎪ ⎪ ⎩ 1 −m(n−1)−a ,
a > 0; −m(n − 1) < a < 0; a < −m(n − 1),
where [a] is the largest integer not exceeding a. Then (i) |λ| ≤ M (a)nm−1 for all H-eigenvalues λ of Hn if m is even; m (ii) |μ| ≤ M (a)n 2 for all Z-eigenvalues μ of Hn . Proof. Using the proof technique of Proposition 1.1, we have |i1 + i2 + · · · + im − m + a| ≥ a for a > 0, |i1 + i2 + · · · + im − m + a| ≥ min{a − [a], 1 − (a − [a])} for − m(n − 1) < a < 0, |i1 + i2 + · · · + im − m + a| ≥ −m(n − 1) − a for a < −m(n − 1).
W. Mei, Y. Song / Linear Algebra and its Applications 532 (2017) 8–24
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In conclusion, 1 ≤ M (a). |i1 + i2 + · · · + im − m + a| Then for each nonzero vector x ∈ Rn , we have n x x · · · x i i i 1 2 m m |Hn x | = i + i + · · · + i − m + a 2 m i1 ,i2 ,··· ,im =1 1 n
≤
i1 ,i2 ,··· ,im
|xi1 xi2 · · · xim | |i + i + · · · + im − m + a| 1 2 =1 n
≤ M (a)
|xi1 ||xi2 | · · · , |xim |
i1 ,i2 ,··· ,im =1 n |xi |)m = M (a)( i=1
= M (a)xm 1 . 1
(i) By the fact x1 ≤ n1− m xm , we obtain 1
1− m |Hn xm | ≤ M (a)xm xm )m 1 ≤ M (a)(n
= M (a)nm−1 xm m, and hence, we have for all nonzero vector x ∈ Rn m |Hn xm | x Hn = ≤ M (a)nm−1 . xm xm m
(3.5)
Let λ be an H-eigenvalues of Hn . Then there exists a nonzero vector y ∈ Rn such that Hn y m−1 = λy [m−1] , and so, Hn y m = y (Hn y m−1 ) = λy y [m−1] = λ
n
yim = λym m
i=1
since m is even. Thus, it follows from (3.5) that m y |Hn y m | ≤ M (a)nm−1 . |λ| = = Hn ym y m m Then (i) follows since λ is a H-eigenvalue.
W. Mei, Y. Song / Linear Algebra and its Applications 532 (2017) 8–24
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(ii) Similarly, we have by the fact x1 ≤
√
nx2 m
m 2 |Hn xm | ≤ M (a)xm 1 ≤ M (a)n x2 ,
and hence, we have for all nonzero vector x ∈ Rn m |Hn xm | m x Hn = ≤ M (a)n 2 . x2 xm 2
(3.6)
Let μ be a Z-eigenvalues of Hn . Then there exists a nonzero vector z ∈ Rn such that Hn z m−1 = μz(z z)
m−2 2
,
and so,
Hn z
m
= z (Hn z
m−1
) = μz z(z z)
m−2 2
=μ
n
m2 zi2
= μzm 2 .
i=1
Thus, by (3.6), we obtain m |Hn z m | z ≤ M (a)n m2 . |μ| = = Hn m z2 z2 Then the desired conclusion follows since μ is a Z-eigenvalue. 2 Let a > 0. Then the generalized Hilbert tensor Hn is positive (all entries are positive) and symmetric, and hence, its spectral radius ρ(Hn ) and its E-spectral radius ρE (Hn ) are its H-eigenvalue and its Z-eigenvalue, respectively. For more details, see Chang, Pearson and Zhang [1,2], Qi [18], Song and Qi [22,23] and Yang and Yang [28,29]. So the following collusions are easy to obtain by Theorem 3.4. Corollary 3.5. Let Hn be an m-order n-dimensional generalized Hilbert tensor. Assume that a > 0. Then (i) |λ| ≤ ρ(Hn ) ≤ (ii) |μ| ≤ ρE (Hn )
nm−1 am ≤ na2
for all eigenvalues λ of Hn if m is even; for all E-eigenvalues μ of Hn .
Remark 3.3. We prove the upper bounds of spectral radii in the above results. These upper bounds may not be the best, then the following problems deserve us further research: What is the best upper bounds for the spectral radii of generalized Hilbert tensors? How to compute such an upper bound?
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Acknowledgements The authors would like to thank the anonymous referees/editors for their valuable suggestions which helped us to improve this manuscript. References [1] K.C. Chang, K. Pearson, T. Zhang, Some variational principles for Z-eigenvalues of nonnegative tensors, Linear Algebra Appl. 438 (2013) 4166–4182. [2] K.C. Chang, K. Pearson, T. Zhang, Perron–Frobenius theorem for nonnegative tensors, Commun. Math. Sci. 6 (2008) 507–520. [3] H. Chen, L. Qi, Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors, J. Ind. Manag. Optim. 11 (4) (2015) 1263–1274. [4] Man-Duen Choi, Tricks for treats with the Hilbert matrix, Amer. Math. Monthly 90 (1983) 301–312. [5] H. Frazer, Note on Hilbert’s inequality, J. Lond. Math. Soc. s1-21 (1) (1946) 7–9. [6] S. Fucik, J. Necas, J. Soucek, V. Soucek, Spectral Analysis of Nonlinear Operators, Lecture Notes in Mathematics, vol. 346, Springer-Verlag, Berlin, Heidelberg, New York, 1973. [7] D. Hilbert, Ein Beitrag zur Theorie des Legendre’schen Polynoms, Acta Math. (Springer Netherlands) 18 (1894) 155–159. [8] J. He, T. Huang, Upper bound for the largest Z-eigenvalue of positive tensors, Appl. Math. Lett. 38 (2014) 110–114. [9] J. He, Bounds for the largest eigenvalue of nonnegative tensors, J. Comput. Anal. Appl. 20 (7) (2016) 1290–1301. [10] J. He, Y.M. Liu, H. Ke, J.K. Tian, X. Li, Bounds for the Z-spectral radius of nonnegative tensors, SpringerPlus 5 (1) (2016) 1727. [11] A.E. Ingham, A note on Hilbert’s inequality, J. Lond. Math. Soc. s1-11 (3) (1936) 237–240. [12] T. Kato, On the Hilbert matrix, Proc. Amer. Math. Soc. 8 (1) (1957) 73–81. [13] W. Li, D. Liu, S.W. Vong, Z-eigenpair bounds for an irreducible nonnegative tensor, Linear Algebra Appl. 483 (2015) 182–199. [14] L.H. Lim, Singular values and eigenvalues of tensors: a variational approach, in: Proc. 1st IEEE International Workshop on Computational Advances of Multi-Tensor Adaptive Processing, Dec. 13–15, 2005, pp. 129–132. [15] W. Magnus, On the spectrum of Hilbert’s matrix, Amer. J. Math. 72 (4) (1950) 699–704. [16] L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput. 40 (2005) 1302–1324. [17] L. Qi, Rank and eigenvalues of a supersymmetric tensor, the multivariate homogeneous polynomial and the algebraic hypersurface it defines, J. Symbolic Comput. 41 (2006) 1309–1327. [18] L. Qi, Symmetric nonnegative tensors and copositive tensors, Linear Algebra Appl. 439 (2013) 228–238. [19] L. Qi, Hankel tensors: associated Hankel matrices and Vandermonde decomposition, Commun. Math. Sci. 13 (1) (2015) 113–125. [20] L. Qi, Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, Society for Industrial & Applied Mathematics, April 19, 2017. [21] Y. Song, L. Qi, Infinite and finite dimensional Hilbert tensors, Linear Algebra Appl. 451 (2014) 1–14. [22] Y. Song, L. Qi, Positive eigenvalue-eigenvector of nonlinear positive mappings, Front. Math. China 9 (1) (2014) 181–199. [23] Y. Song, L. Qi, Spectral properties of positively homogeneous operators induced by higher order tensors, SIAM J. Matrix Anal. Appl. 34 (4) (2013) 1581–1595. [24] Y. Song, L. Qi, Necessary and sufficient conditions for copositive tensors, Linear Multilinear Algebra 63 (1) (2015) 120–131. [25] O. Taussky, A remark concerning the characteristic roots of the finite segments of the Hilbert matrix, Quart. J. Math. os-20 (1) (1949) 80–83. [26] Y. Wang, K. Zhang, H. Sun, Criteria for strong H-tensors, Front. Math. China 11 (3) (2016) 577–592.
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