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Article

A New Seminorm for d-Tuples of A-Bounded Operators and Their Applications

by
Najla Altwaijry
1,*,†,
Kais Feki
2,3,† and
Nicuşor Minculete
4,†
1
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
2
Faculty of Economic Sciences and Management of Mahdia, University of Monastir, Mahdia 5111, Tunisia
3
Laboratory Physics-Mathematics and Applications (LR/13/ES-22), Faculty of Sciences of Sfax, University of Sfax, Sfax 3018, Tunisia
4
Department of Mathematics and Computer Science, Transilvania University of Brasov, 500091 Brasov, Romania
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Mathematics 2023, 11(3), 685; https://doi.org/10.3390/math11030685
Submission received: 27 December 2022 / Revised: 23 January 2023 / Accepted: 24 January 2023 / Published: 29 January 2023
(This article belongs to the Special Issue Mathematical Inequalities, Models and Applications)

Abstract

:
The aim of this paper was to introduce and investigate a new seminorm of operator tuples on a complex Hilbert space H when an additional semi-inner product structure defined by a positive (semi-definite) operator A on H is considered. We prove the equality between this new seminorm and the well-known A-joint seminorm in the case of A-doubly-commuting tuples of A-hyponormal operators. This study is an extension of a well-known result in [Results Math 75, 93(2020)] and allows us to show that the following equalities r A ( T ) = ω A ( T ) = T A hold for every A-doubly-commuting d-tuple of A-hyponormal operators T = ( T 1 , , T d ) . Here, r A ( T ) , T A , and ω A ( T ) denote the A-joint spectral radius, the A-joint operator seminorm, and the A-joint numerical radius of T , respectively.

1. Introduction

In functional analyses, many authors have studied the tuples of operators. For example, we refer to [1,2,3,4,5] and the references therein.
Consider a complex Hilbert space ( H , · , · ) , where the norm induced by · , · is denoted by · . The set B ( H ) denotes the C * -algebra of all bounded linear operators acting on H with identity I H (or shortly I). If H is n-dimensional, we identify B ( H ) with the space M n of all n × n matrices with entries in the complex field and denote its identity by I n . In what follows, by an operator, we mean a bounded linear operator. We will mention some specific notions of an operator, i.e., the null space of every operator T is denoted by N ( T ) , its range by R ( T ) , and T * is the adjoint of T. An operator T B ( H ) is said to be positive if T x , x 0 for all x H . We write T 0 if T is positive. If T 0 , then T 1 / 2 means the square root of T. The commutator of two operators T , S B ( H ) is defined as [ T , S ] : = T S S T . It is easy to see that [ T λ I , S μ I ] = [ T , S ] , for every λ , μ C and T , S B ( H ) . Recall that T B ( H ) is called normal (respectively hyponormal) if [ T * , T ] = 0 (respectively, [ T * , T ] 0 ).
Next, we present some inequalities related to operators that we need in the future. First, we give the classical Schwarz inequality for a positive operator T B ( H ) :
| T x , y | 2 T x , x T y , y ,
where x , y H .
In [6], Halmos obtains a result similar to the inequality above
T x , x T x , x 1 / 2 T * x , x 1 / 2
for every T B ( H ) and for any x , y H . In [7], Kato proves a Schwarz-type inequality (1), which generalizes the inequality of Halmos:
| T x , y | 2 | T | 2 θ x , x | T * | 2 ( 1 θ ) y , y
for all operators T B ( H ) , for every vector x , y H , and θ [ 0 , 1 ] . McCarthy [8] gives an important inequality in the theory of operators as follows:
Lemma 1 
(Theorem 1.4 in [8]). Let T B ( H ) be a positive operator and x H satisfy x = 1 . Then, for r 1 ,
T x , x r T r x , x .
For 0 r 1 , the above inequality is reversed.
In what follows, we assume that A is a positive nonzero operator that defines the following positive semi-definite sesquilinear form:
· , · A : H × H C ( x , y ) x , y A : = A x , y = A 1 / 2 x , A 1 / 2 y .
The seminorm induced by · , · A is given by x A = x , x A , x H . It can be seen that · A is a norm on H if and only if A is injective, and the semi-Hilbert space ( H , · A ) is complete if and only if R ( A ) is closed in H .
Definition 1 
([9]). An operator S B ( H ) is called an A-adjoint of T B ( H ) , if we have T x , y A = x , S y A ( A S = T * A ) for every x , y H .
The existence of an A-adjoint operator is not guaranteed. Thus, we denote by B A ( H ) the set of all operators that admit A-adjoints. Using Douglas’ theorem [10], we obtain the following:
B A ( H ) = T B ( H ) ; R ( T * A ) R ( A )
and
B A 1 / 2 ( H ) = T B ( H ) ; c > 0 ; T x A c x A , x H .
When T B A 1 / 2 ( H ) , we say that T is A-bounded. The sets B A ( H ) and B A 1 / 2 ( H ) are subalgebras of B ( H ) , which are neither closed nor dense in B ( H ) . Moreover, the inclusions
B A ( H ) B A 1 / 2 ( H ) B ( H )
hold. We have equality if A is injective and R ( A ) ¯ = R ( A ) , where R ( A ) ¯ means the closure of R ( A ) in the norm topology of H (see [11]). Further, · , · A gives the following seminorm on B A 1 / 2 ( H )
T A : = sup x R ( A ) ¯ , x 0 T x A x A = sup x H , x A = 1 T x A = sup x , y H , x A = y A = 1 | T x , y A | <
(see [12] and the references therein). It is useful to note that if T B A 1 / 2 ( H ) , then T ( N ( A ) ) N ( A ) . Further, T S A T A S A for any T , S B A 1 / 2 ( H ) . Let X denote the Moore–Penrose pseudo-inverse of an operator X (for more details concerning this operator, see [11]). Following [11], we have: T B A 1 / 2 ( H ) implies that A 1 / 2 T ( A 1 / 2 ) B ( H ) and
T A = A 1 / 2 T ( A 1 / 2 ) .
In 2012, Saddi [13] introduced the A-numerical radius of an operator T B ( H ) by
ω A ( T ) : = sup | T x , x A | ; x H , x A = 1 .
In 2020, the concept of the A-spectral radius of A-bounded operators was defined in [14] as follows:
r A ( T ) : = inf n 1 T n A 1 n = lim n T n A 1 n .
Note that T A and ω A ( T ) may equal + for some T B ( H ) (see [14]). However, the following relation shows that · A and ω A ( · ) are equivalent seminorms on B A 1 / 2 ( H ) :
r A ( T ) max 1 2 T A , r A ( T ) ω A ( T ) T A .
For the proof of (6), we refer to the following references [12,14]. If A = I , then the classical definitions of the operator norm, numerical radius, and spectral radius for Hilbert space operators are obtained and are simply denoted by · , ω ( · ) and r ( · ) .
If T B A ( H ) , then by Douglas’s theorem [10] there exists a unique solution, given by T A , of the following problem
A X = T * A , R ( X ) R ( A ) ¯ .
Note that T A = A T * A , where A is the Moore–Penrose pseudo-inverse of A (see [11]). If T , S B A ( H ) , then T S , α T + β S B A ( H ) for every α , β R and we have ( T S ) A = S A T A and ( α T + β S ) A = α ¯ T A + β ¯ S A . Moreover, if P R ( A ) ¯ denotes the orthogonal projection onto R ( A ) ¯ , then for a given T B A ( H ) , we have T A B A ( H ) , ( T A ) A = P R ( A ) ¯ T P R ( A ) ¯ and ( ( T A ) A ) A = T A . For more details about the operator T A , one can see [9,11,15]. Furthermore, we recall that an operator T is said to be A-positive if A T is a positive operator and we write T A 0 . It can be observed that A-positive operators are in B A ( H ) . For T , S B ( H ) , the notation T A S means T S A 0 . When A = I , then T I S will simply be denoted by T S .
The structure of this paper is organized as follows: in Section 2, we give some notions that characterize a d-tuple of operators T = ( T 1 , , T d ) B ( H ) d . In Section 3, we introduce a new joint norm of tuples of operators that generalizes the joint norm given in (12) and define the class of doubly-commuting tuples of hyponormal operators acting on an A-weighted Hilbert space, where A is a positive operator that is not assumed to be invertible. We proved a generalization of the well-known result due to G. Popescu [16]. We also present an inequality that characterizes the Euclidean norm of an operator tuple T = ( T 1 , , T d ) B ( H ) d . In Section 4, we give several characterizations related to the operators from B A 1 / 2 ( H ) and the operators from B ( R ( A 1 / 2 ) ) . For an A-doubly-commuting d-tuple of hyponormal operators, we prove the equalities T e , A = T A and r A ( T ) = T A = ω A ( T ) . The motivation for our investigation comes from a recent paper [17].

2. Preliminaries

To prepare the framework in which we will work, we present in this section some notions and notations that will be useful in this paper.
Let N and N * denote the set of nonnegative and positive integers, respectively. Let d N * and T = ( T 1 , , T d ) B ( H ) d be a d-tuple of operators. If [ T i , T j ] = 0 for all i , j { 1 , , d } , then T is said to be a commuting tuple. Moreover, if T is a commuting d-tuple of operators and [ T i * , T j ] = 0 for every 1 i j d , then it is called a doubly-commuting operator tuple.
In the next definition, we recall two important classes of operators in semi-Hilbert spaces.
Definition 2 
([14]). An operator T B A ( H ) is called
(i) 
A-normal if [ T A , T ] = 0 ;
(ii) 
A-hyponormal if [ T A , T ] A 0 .
For some results concerning the above two classes of operators, see [14] and the references therein. For T B A ( H ) , the equalities
r A ( T ) = ω A ( T ) = T A
hold for the class of A-normal, A-hyponormal, and A-positive operators (see [14]). Since T A T A 0 and T T A A 0 , then an application of the second equality in (7) together with the last equality in (3) shows that
T A T A = T T A A = T A 2 = T A A 2 .
Now, associated with a d-tuple of operators, T = ( T 1 , , T d ) B ( H ) d (not necessarily commuting), the following quantities
T A : = sup j = 1 d T j x A 2 ; x H , x A = 1 ,
and
ω A ( T ) : = sup j = 1 d | T j x , x A | 2 ; x H , x A = 1
are defined in [12]. If T j B A 1 / 2 ( H ) for all j { 1 , , d } , then one can verify that · A and ω A ( · ) two seminorms on B A 1 / 2 ( H ) d . Notice that ω A ( T ) and T A are called the A-joint numerical radius and the A-joint operator seminorm of T , respectively.
In [18], H. Baklouti et al. introduced the concept of the A-joint spectral radius associated with a d-tuple of commuting operators T = ( T 1 , , T d ) B A ( H ) d as follows
r A ( T ) : = inf n N * | α | = n , α N d n ! α ! T A α T α A 1 2 n = lim n | α | = n , α N d n ! α ! T A α T α A 1 2 n ,
where T A = ( T 1 A , , T d A ) . Moreover, for the multi-index α = ( α 1 , , α d ) N d , we will use the following notations:
T α : = k = 1 d T k α k , | α | : = j = 1 d | α j | and α ! : = k = 1 d α k ! .
We mention here that the second equality in (9) has also been proved by Baklouti et al. in [18]. Notice that for every commuting operator tuple T = ( T 1 , , T d ) B A ( H ) d , we have
r A ( T ) max 1 2 d T A , r A ( T ) ω A ( T ) T A
(see Theorem 2.4 in [12] and Theorem 2.2 in [19]). In [19], it is stated that if T = ( T 1 , , T d ) B A ( H ) d is any d-tuple of commuting A-normal operators, then
r A ( T ) = ω A ( T ) = T A .
One of the main targets of this work is to establish the equalities in (11) for a new class of multivariable operators.
Next, for A = I , we define r I ( T ) , ω I ( T ) , and T I which will simply be denoted by r ( T ) , ω ( T ) and T , respectively. Thus, we obtain
T : = sup j = 1 d T j x 2 ; x H , x = 1 ,
and
ω ( T ) : = sup j = 1 d | T j x , x | 2 ; x H , x = 1 .
The last equality is given in [20] by M. Chō and M. Takaguchi and is the Euclidean operator radius of an operator tuple T = ( T 1 , , T d ) B ( H ) d , see also [16].
For T = ( T 1 , , T d ) B ( H ) d , G. Popescu defined in [16] the following quantity
T e : = sup ( λ 1 , , λ d ) B d λ 1 T 1 + + λ d T d ,
where B d denotes the open unit ball of C d with respect to the Euclidean norm, i.e.,
B d : = λ = ( λ 1 , , λ d ) C d ; λ 2 2 : = j = 1 d | λ j | 2 < 1 .
It is clear that we can change B d with its closure in (12) without changing the value of T e . Note that · e defines a norm on B ( H ) d . Moreover, in [17], the following equality is established:
T = T e
for every doubly-commuting d-tuple of hyponormal operators T . It is important to mention that G. Popescu proved in [16] that the following inequalities hold
1 d j = 1 d T j T j * T e j = 1 d T j T j *
for any d-tuple T = ( T 1 , , T d ) B ( H ) d . Furthermore, it has been shown in [16] that the constants 1 d and 1 are the best choices possible.

3. New Joint Seminorm for Operator Tuples

In this section, we aim to introduce and investigate a new joint seminorm for d-tuples of A-bounded operators. An alternative and easy proof of a well-known result due to G. Popescu [16] is established.
First, we introduce the following definition, which is a natural generalization of (12).
Definition 3. 
Let T = ( T 1 , , T d ) B A 1 / 2 ( H ) d . The A-Euclidean seminorm of the d-tuple of A-bounded operators T is given by
T e , A : = sup ( λ 1 , , λ d ) B d λ 1 T 1 + + λ d T d A .
In the next proposition, we state some connections between the seminorms · e , A and · A .
Proposition 1. 
Let T = ( T 1 , , T d ) B A 1 / 2 ( H ) d be the d-tuple of the operators. Then, the following assertions hold:
(1) 
T e , A T A ;
(2) 
If T k B A ( H ) for all k { 1 , , d } , then T A e , A = T e , A and
1 d max T A , T A A T e , A ,
where T A = ( T 1 A , , T d A ) .
Proof. 
(1) Let x H and λ = ( λ 1 , , λ d ) B d . Then, by applying the Cauchy–Schwarz inequality (in short (C–S)) and making several calculations, we deduce that
j = 1 d λ j T j x A 2 = j = 1 d λ j T j x , k = 1 d λ k T k x A = j = 1 d k = 1 d λ j λ k ¯ T j x , T k x A j = 1 d k = 1 d | λ j | × | λ k | × T j x A T k x A = j = 1 d | λ j | × T j x A 2 .
By applying the inequality (C–S) again, we obtain the following inequality
k = 1 d λ k T k x A 2 λ 2 2 j = 1 d T j x A 2 .
Then, by taking the supremum over all x H with x A = 1 , we find
k = 1 d λ k T k A λ 2 T A .
So, the desired inequality is proved by taking the supremum over all λ B d .
(2) The fact that T A e , A = T e , A follows trivially since X A A = X A for all X B A ( H ) . Now, in order to prove (15), we need to recall from [16] the following facts: if we denote by S d the unit sphere of C d and σ the rotation-invariant positive Borel measure on S d for which σ ( S d ) = 1 , then for all μ = ( μ 1 , , μ d ) C d , we have
S d | μ k | 2 d σ ( μ ) = 1 d , k { 1 , , d } and S d μ i μ j ¯ d σ ( μ ) = 0 , 1 i j d .
Now, let B ¯ d denote the closed unit ball of C d . It is clear that
T e , A 2 = sup ( λ 1 , , λ d ) B d j = 1 d λ j T j A 2 = sup ( λ 1 , , λ d ) B ¯ d j = 1 d λ j T j A 2 .
Further, by using (8), we see that
T e , A 2 = sup ( λ 1 , , λ d ) B ¯ d j = 1 d λ j T j A j = 1 d λ j T j A = sup ( λ 1 , , λ d ) B ¯ d sup x A = 1 j = 1 d λ j ¯ T j A j = 1 d λ j T j x , x A = sup x A = 1 sup ( λ 1 , , λ d ) B ¯ d i , j = 1 d λ i ¯ λ j T i A T j x , x A .
On the other hand, since σ ( S d ) = 1 , then it follows that
sup ( λ 1 , , λ d ) B ¯ d i , j = 1 d λ i ¯ λ j T i A T j x , x A = S d sup ( λ 1 , , λ d ) B ¯ d i , j = 1 d λ i ¯ λ j T i A T j x , x A d σ ( μ ) S d i , j = 1 d μ i ¯ μ j T i A T j x , x A d σ ( μ ) ,
for all x H . This implies that, through (16),
T e , A 2 sup x A = 1 S d i , j = 1 d μ i μ j ¯ T i A T j x , x A d σ ( μ ) = 1 d sup x A = 1 i = 1 d T i A T i x , x A = 1 d T A 2 .
This proves that
1 d T A T e , A .
By replacing T k by T k A in (17) and then using the fact that T A e , A = T e , A , we have
1 d T A A T e , A .
Combining (17) together with (18) yields (15) as desired. □
Remark 1. 
(1) If T = ( T 1 , , T d ) B A ( H ) d , then clearly k = 1 d T k A T k A 0 . Hence, a direct application of (7) shows that
T A = j = 1 d T j A T j A .
It should be mentioned here that the equality T A A = T A may not be correct even if T is a commuting operator tuple. Indeed, let us consider the following matrices in M 3 : A = 1 0 0 0 2 0 0 0 1 , T 1 = 0 1 0 0 0 0 0 0 0 and T 2 = 0 0 1 0 0 0 0 0 0 . We remark that [ T 1 , T 2 ] = 0 . Furthermore, by using the fact that T k A = A T k * A with k { 1 , 2 } , it can be seen that T 1 A = 0 0 0 1 2 0 0 0 0 0 and T 2 A = 0 0 0 0 0 0 1 0 0 . Further, by applying (19) and (4), we can show that
( T 1 , T 2 ) A = T 1 A T 1 + T 2 A T 2 A = 1
and
( T 1 A , T 2 A ) A = ( T 1 A ) A T 1 A + ( T 2 A ) A T 2 A A = 5 2 .
(2) In virtue of Proposition 1, we infer that · A and · e , A are equivalent seminorms on B A ( H ) d .
The following corollary provides a generalization and improvement of the well-known result due to G. Popescu [16].
Corollary 1. 
Let T = ( T 1 , , T d ) B A ( H ) d be a d-tuple of operators. Then, the inequality
1 d max α , β T e , A min α , β
holds, where α = k = 1 d T k T k A A and β = k = 1 d T k A T k A .
Proof. 
By applying Proposition 1 together with (19), we deduce that
1 d k = 1 d T k A T k A T e , A k = 1 d T k A T k A .
By replacing T k by T k A in (21), we can see that
1 d k = 1 d T k T k A A A T A e , A k = 1 d T k T k A A A ,
from which we have
1 d k = 1 d T k T k A A T e , A k = 1 d T k T k A A .
A combination of (21) together with (22) yields (20) as desired. □
Remark 2. 
Note that the following equality
j = 1 d T j A T j A = j = 1 d T j T j A A
may not be correct for some d-tuple of operators T = ( T 1 , , T d ) B A ( H ) d even if A = I . Indeed, we consider the following matrices in M 2 : A = I 2 , T 1 = 1 0 0 0 and T 2 = 0 0 1 0 . It is not difficult to check that
j = 1 2 T j T j * = 1 2 = j = 1 2 T j * T j .
In the next theorem, we give a new formula of T A , e for T B A 1 / 2 ( H ) d , which allows us to prove that · A , e and · A are two equivalent seminorms on B A 1 / 2 ( H ) d . Notice that our new techniques provide an alternative and easy proof of the inequalities (14), which were first proved in [16].
Theorem 1. 
Let T = ( T 1 , , T d ) B A 1 / 2 ( H ) d . Then, the equality
T A , e = sup j = 1 d | T j x , y A | 2 ; x , y H , x A = y A = 1
holds.
Proof. 
By using (3), we see that
T e , A = sup ( λ 1 , , λ d ) B d λ 1 T 1 + + λ d T d A = sup ( λ 1 , , λ d ) B d sup k = 1 d λ k T k x A ; x H , x A = 1 = sup sup ( λ 1 , , λ d ) B d k = 1 d λ k T k x A ; x H , x A = 1 .
Moreover, recall from [19] that for complex numbers z 1 , , z d , we have
sup ( λ 1 , , λ d ) B d k = 1 d λ k z k = k = 1 d | z k | 2 .
Now, let x , y H . By using (25), we have
j = 1 d | T j x , y A | 2 = sup ( λ 1 , , λ d ) B d j = 1 d λ j T j x , y A = sup ( λ 1 , , λ d ) B d j = 1 d λ j T j x , y A
Hence, by taking the supremum over all y H with y A = 1 in the last equality we have
sup y H , y A = 1 k = 1 d | T k x , y A | 2 = sup y H , y A = 1 sup ( λ 1 , , λ d ) B d k = 1 d λ k T k x , y A .
This yields that
sup y H , y A = 1 k = 1 d | T k x , y A | 2 = sup ( λ 1 , , λ d ) B d sup y H , y A = 1 k = 1 d λ k T k x , y A .
On the other hand, it is not difficult to check that
sup | u , v A | ; v H , v A = 1 = u A , u H .
Thus, by using (26) and (27), we obtain
sup y H , y A = 1 k = 1 d | T k x , y A | 2 = sup ( λ 1 , , λ d ) B d k = 1 d λ k T k x A .
Combining (28) together with (24) yields (23) as required, and, hence, the proof is complete. □
Remark 3. 
By letting A = I in (23), we obtain a well-known result established by Dragomir in Theorem 9 in [21], and when the 2-tuple is ( T , T A ) , where T B A ( H ) , we obtain a recent result in [22].
The following corollary is an application of Theorem 1 and provides an improvement of the results given in Proposition 1 since B A ( H ) B A 1 / 2 ( H ) . Moreover, the new Formula (23) enables us to derive an alternative and easy proof of the inequalities (14).
Corollary 2. 
Let T = ( T 1 , , T d ) B A 1 / 2 ( H ) d be a d-tuple of operators. Then,
1 d T A T e , A T A .
Proof. 
By using (23) and then applying the inequality (C–S), we easily prove the second inequality in (29). Now, let x H be such that x A = 1 . Assume that T k x N ( A ) for all k { 1 , , d } and let
y k = T k x T k x A , k { 1 , , d } .
(If T k 0 x N ( A ) for some k 0 { 1 , , d } , we choose y k 0 = x ). We clearly have
y k A = 1 and | T k x , y k A | 2 = T k x A 2 , k { 1 , , d } .
Thus, by applying (23), we have
T A , e 2 k = 1 d | T k x , y k A | 2 | T 1 x , y 1 A | 2 = T 1 x A 2 .
Similarly, we prove that T A , e 2 T i x A 2 for all i { 1 , , d } . This yields
d T A , e 2 k = 1 d T k x A 2 .
Therefore, by taking the supremum over all x H with x A = 1 in the last inequality, we have
T A , e 1 d T A .
Hence, the proof is complete. □
Remark 4. 
By letting A = I in (29) and then replacing T k with T k * for all k { 1 , , d } we easily obtain the inequalities (14) that have been already established by G. Popescu in [16] by using a different argument.
To establish our next result, we require the following lemma.
Lemma 2. 
For any vectors x 1 , x 2 , , x d in H and for arbitrary complex numbers λ 1 , λ 2 , , λ d , with λ i 0 , i = 1 , d ¯ , we have
i = 1 d x i A 2 i = 1 d λ i x i A 2 i = 1 d | λ i | 2 + max i , j { 1 , , d } λ i ¯ x j λ j ¯ x i A 2 | λ i | 2 + | λ j | 2
for any d 2 .
Proof. 
We use, as in [23] or [24], the technique of the monotony of a sequence. Consider the sequence
S d = i = 1 d x i A 2 i = 1 d λ i x i A 2 i = 1 d | λ i | 2 , d 1 .
By studying the monotony of sequence S k , k d , we have
S k + 1 S k = x k + 1 A 2 + i = 1 k λ i x i A 2 i = 1 k | λ i | 2 i = 1 k λ i x i + λ k + 1 x k + 1 A 2 i = 1 k | λ i | 2 + | λ k + 1 | 2 .
For two vectors x , y H and for complex numbers λ , μ 0 , the following equality holds:
x A 2 | λ | 2 + y A 2 | μ | 2 x + y A 2 | λ | 2 + | μ | 2 = | μ | 2 x | λ | 2 y A 2 | λ | 2 | μ | 2 ( | λ | 2 + | μ | 2 ) .
Since the term on the right side of equality (30) is positive, then we have
x A 2 | λ | 2 + y A 2 | μ | 2 x + y A 2 | λ | 2 + | μ | 2 .
Now, using the inequality from (31), we have:
x k + 1 A 2 + i = 1 k λ i x i A 2 i = 1 k | λ i | 2 = λ k + 1 x k + 1 A 2 | λ k + 1 | 2 + i = 1 k λ i x i A 2 i = 1 k | λ i | 2 i = 1 k λ i x i + λ k + 1 x k + 1 A 2 i = 1 k | λ i | 2 + | λ k + 1 | 2 .
It is easy to see that S k + 1 S k 0 , that is, the sequence S k is increasing. Therefore, we deduce that
S d S d 1 S 2 S 1 = 0 .
However, by applying relation (30) for x = λ 1 x 1 , y = λ 2 x 2 , λ = λ 1 and μ = λ 2 , we obtain
S 2 = x 1 A 2 + x 2 A 2 λ 1 x 1 + λ 2 x 2 A 2 | λ 1 | 2 + | λ 2 | 2 = λ 1 ¯ x 2 λ 2 ¯ x 1 A 2 | λ 1 | 2 + | λ 2 | 2 .
Taking into account that we can rearrange the terms of the two sequences, we obtain the inequality:
S d = i = 1 d x i A 2 i = 1 d λ i x i A 2 i = 1 d | λ i | 2 S 2 = λ 1 ¯ x 2 λ 2 ¯ x 1 A 2 | λ 1 | 2 + | λ 2 | 2 .
Consequently, we deduce the inequality of the statement. □
We are now able to establish the following result.
Theorem 2. 
Let T = ( T 1 , , T d ) B A 1 / 2 ( H ) d be a d-tuple of operators. Then, the inequality
T A T e , A + max i , j { 1 , , d } sup ( λ 1 , , λ d ) B d inf x A = 1 ( λ i ¯ T j λ j ¯ T i ) x A 2
holds for any d 2 .
Proof. 
In Lemma 2, set x i = T i x for all i { 1 , , d } , then
i = 1 d T i x A 2 i = 1 d λ i T i x A 2 i = 1 d | λ i | 2 + max i , j { 1 , , d } λ i ¯ T j x λ j ¯ T i x A 2 | λ i | 2 + | λ j | 2 .
First, we take the supremum over all x H with x A = 1 in relation (33), we deduce
T A i = 1 d λ i T i A 2 i = 1 d | λ i | 2 + max i , j { 1 , , d } inf x A = 1 ( λ i ¯ T j λ j ¯ T i ) x A 2 | λ i | 2 + | λ j | 2 .
Therefore, it we take the supremum over all ( λ 1 , , λ d ) B d in relation (34), then we find the inequality of the statement. □
Remark 5. 
By letting A = I in (32), we obtain
T T e + max i , j { 1 , , d } sup ( λ 1 , , λ d ) B d inf x = 1 ( λ i ¯ T j λ j ¯ T i ) x 2
for any d-tuple of operators T = ( T 1 , , T d ) B ( H ) d and d 2 .
Next, we will present a result that characterizes the Euclidean norm of an operator tuple T = ( T 1 , , T d ) B ( H ) d .
Proposition 2. 
Let T = ( T 1 , , T d ) B ( H ) d be a d-tuple of operators. The following inequality holds:
T e T θ T * 1 θ ,
where θ [ 0 , 1 ] . Here T * = ( T 1 * , , T d * ) .
Proof. 
First, we will prove a radon-type inequality,
k = 1 d a k θ b k 1 θ k = 1 d a k θ k = 1 d b k 1 θ ,
for every a k 0 and b k > 0 with k { 1 , , d } . If we apply the Jensen inequality for the function f ( x ) = x θ , which is concave for θ [ 0 , 1 ] , we deduce
k = 1 d b k a k b k θ k = 1 d b k k = 1 d b k a k b k k = 1 d b k θ ,
which is equivalent to inequality (36). In [25], Dragomir applied Hölder’s inequality for this. For θ 1 , the function f ( x ) = x θ is convex and the inequality sign in (36) is flipped, obtaining the classical Radon inequality
k = 1 d a k θ b k θ 1 k = 1 d a k θ k = 1 d b k θ 1 .
Thus, we have
k = 1 d | T k x , y | 2 K a t o k = 1 d | T k | 2 θ x , x | T k * | 2 ( 1 θ ) y , y M c C a r t h y k = 1 d | T k | 2 x , x θ | T k * | 2 y , y 1 θ ( 36 ) k = 1 d | T k | 2 x , x θ k = 1 d | T k * | 2 y , y 1 θ = k = 1 d T k x 2 1 / 2 2 θ k = 1 d T k * y 2 1 / 2 2 ( 1 θ ) .
Therefore, we deduce
k = 1 d | T k x , y | 2 1 / 2 k = 1 d T k x 2 1 / 2 θ k = 1 d T k * y 2 1 / 2 1 θ .
Consequently, by taking the supremum over x , y H with x = y = 1 in inequality (37) and taking into account the equality from Theorem 1 for A = I , we obtain the desired result. □
Remark 6. 
If we take θ = 1 in relation (35), then T e T and for θ = 0 in the same relation, we obtain T e T * . Using the Kittaneh–Manasrah inequality [26] and inequality (35), we found the following inequality:
T e θ T + ( 1 θ ) T * min { θ , 1 θ } T T * 2
for all T = ( T 1 , , T d ) B ( H ) d a d-tuple of positive operators.

4. A -Doubly-Commuting Tuples of A -Hyponormal Operators

In this section, we give several characterizations related to the operators from B A 1 / 2 ( H ) and the operators from B ( R ( A 1 / 2 ) ) . For an A-doubly-commuting d-tuple of hyponormal operators, we proved the equalities T e , A = T A and r A ( T ) = T A = ω A ( T ) .
Let us introduce the following definition.
Definition 4. 
Let T = ( T 1 , , T d ) B A ( H ) d . The d-tuple T is said to be A-doubly commuting if:
(i) 
it is commuting, i.e., [ T i , T k ] = 0 for all i , k { 1 , , d } ,
(ii) 
T i A T k = T k T i A for all 1 i k d .
In this section, we will study the connection between T A and T e , A , when T is a d-tuple of A-doubly-commuting tuples of A-hyponormal operators. For this purpose, we need to recall some aspects: the semi-inner product · , · A induces an inner product on the quotient space H / N ( A ) is given by
( x ¯ , y ¯ ) = A x , y ,
for any x ¯ = x + N ( A ) , y ¯ = y + N ( A ) H / N ( A ) . We remark that H / N ( A ) , ( · , · ) is not complete unless R ( A ) is closed in H . However, L. de Branges and J. Rovnyak [27] proved that the completion of H / N ( A ) is isometrically isomorphic to the Hilbert space R ( A 1 / 2 ) : = R ( A 1 / 2 ) , · , · R ( A 1 / 2 ) , where · , · R ( A 1 / 2 ) is given by
A 1 / 2 δ , A 1 / 2 ξ R ( A 1 / 2 ) : = P R ( A ) ¯ δ , P R ( A ) ¯ ξ , δ , ξ H .
It is obvious that · R ( A 1 / 2 ) stands for the norm induced by the inner product · , · R ( A 1 / 2 ) . We mention here that, in view of Proposition 2.1. in [15], we have R ( A ) is dense in R ( A 1 / 2 ) . Further, since R ( A ) R ( A 1 / 2 ) , then we observe that
A δ , A ξ R ( A 1 / 2 ) = δ , ξ A , δ , ξ H ,
which implies that
A ξ R ( A 1 / 2 ) = ξ A , x H .
In the next proposition, we give an interesting connection between operators in B A 1 / 2 ( H ) and operators in B ( R ( A 1 / 2 ) ) . Furthermore, we summarize in the same proposition some useful properties.
Proposition 3 
([14,15,19,28]). Let T B ( H ) . Then there exists T ˜ B ( R ( A 1 / 2 ) ) such that T ˜ Z A = Z A T if and only if T B A 1 / 2 ( H ) . In such cases T ˜ is unique. Here Z A : H R ( A 1 / 2 ) is defined by Z A x = A x . Furthermore, for every T , S B A 1 / 2 ( H ) and α , β C , we have the following properties:
(1) 
T A = T ˜ B ( R ( A 1 / 2 ) ) and r A ( T ) = r ( T ˜ ) .
(2) 
If T B A ( H ) , then T A ˜ = ( T ˜ ) * and ( T A ) A ˜ = T ˜ .
(3) 
If T k B A 1 / 2 ( H ) and η k C for all k { 1 , , d } , then we have
(i) 
k = 1 d η k T k B ( R ( A 1 / 2 ) ) and k = 1 d η k T k ˜ = k = 1 d η k T k ˜ ;
(ii) 
k = 1 d η k T k B ( R ( A 1 / 2 ) ) and k = 1 d η k T k ˜ = k = 1 d η k T k ˜ .
(4) 
If T = ( T 1 , , T d ) B A ( H ) d such that T i T j = T j T i for all i , j { 1 , , d } , then r A ( T ) = r ( T ˜ ) , where T ˜ = ( T 1 ˜ , , T d ˜ ) .
The following lemma is also useful in proving our results in this section.
Lemma 3. 
Let T , S B A 1 / 2 ( H ) . Then, the following assertions hold:
(i) 
If T A S , then T ˜ S ˜ ;
(ii) 
If T B A ( H ) is A-hyponormal, then T ˜ is hyponormal on R ( A 1 / 2 ) .
Proof. 
(1) Let x H . By applying (38) together with Proposition 3 (3), we have
( T S ) x , x A = A ( T S ) x , A x R ( A 1 / 2 ) = ( T S ) ˜ A x , A x R ( A 1 / 2 ) = ( T ˜ S ˜ ) A x , A x R ( A 1 / 2 ) 0 ,
where the last inequality follows by using the fact that T A S . Now, by using the fact that R ( A ) is dense in R ( A 1 / 2 ) , we can check that
( T ˜ S ˜ ) A 1 / 2 x , A 1 / 2 x R ( A 1 / 2 ) 0 , x H .
Hence, T ˜ S ˜ is a positive operator on the Hilbert space R ( A 1 / 2 ) . Therefore, T ˜ S ˜ as required.
(2) Since T is A-hyponormal, then T A T A T T A . Hence, by the first assertion of this lemma, we deduce that T A T ˜ T T A ˜ . By Proposition 3, this implies that ( T ˜ ) * T ˜ T ˜ ( T ˜ ) * . Therefore, T ˜ is an hyponormal operator on R ( A 1 / 2 ) . □
Now, we are able to prove the following result.
Theorem 3. 
Let T = ( T 1 , , T d ) B A ( H ) d be an A-doubly-commuting d-tuple of A-hyponormal operators. Then, the following equality
T e , A = T A
holds.
Proof. 
Since T = ( T 1 , , T d ) B A ( H ) d B A 1 / 2 ( H ) d , then by applying Proposition 3 we deduce that for each k { 1 , , d } there exists T k ˜ B ( R ( A 1 / 2 ) ) such that Z A T k = T k ˜ Z A . Another application of Proposition 3 shows that
T e , A = sup ( λ 1 , , λ d ) B d λ 1 T 1 + + λ d T d A = sup ( λ 1 , , λ d ) B d λ 1 T 1 + + λ d T d ˜ B ( R ( A 1 / 2 ) ) = sup ( λ 1 , , λ d ) B d λ 1 T 1 ˜ + + λ d T d ˜ B ( R ( A 1 / 2 ) ) ,
and hence,
T e , A = T ˜ e ,
where T ˜ = ( T 1 ˜ , , T d ˜ ) . On the other hand, we observe that the A-joint seminorm of T can be written as: T A = sup { λ 2 ; λ = ( λ 1 , , λ d ) Ω A ( T ) } where
Ω A ( T ) : = T 1 x A , , T d x A ; x H , x A = 1 .
If A = I , we simply denote Ω I ( · ) by Ω ( · ) . In particular, we observe that
Ω ( T ˜ ) = ( T 1 ˜ y R ( A 1 / 2 ) , , T d ˜ y R ( A 1 / 2 ) ) ; y R ( A 1 / 2 ) , y R ( A 1 / 2 ) = 1 .
Now, by using the decomposition H = N ( A ) R ( A 1 / 2 ) ¯ together with (39), we obtain
Ω A ( T ) = T 1 x A , , T d x A ; x R ( A 1 / 2 ) ¯ ; x A = 1 = ( A T 1 x R ( A 1 / 2 ) , , A T d x R ( A 1 / 2 ) ) ; x R ( A 1 / 2 ) ¯ , A x R ( A 1 / 2 ) = 1 ,
from which
Ω A ( T ) = ( T 1 ˜ A x R ( A 1 / 2 ) , , T d ˜ A x R ( A 1 / 2 ) ) ; x R ( A 1 / 2 ) ¯ , A x R ( A 1 / 2 ) = 1 .
This immediately implies that
Ω A ( T ) Ω ( T ˜ )
Furthermore, it can be seen that
Ω ( T ˜ ) = T 1 ˜ A 1 / 2 x R ( A 1 / 2 ) , , T d ˜ A 1 / 2 x R ( A 1 / 2 ) ; x H , A 1 / 2 x R ( A 1 / 2 ) = 1 = T 1 ˜ A 1 / 2 x R ( A 1 / 2 ) , , T d ˜ A 1 / 2 x R ( A 1 / 2 ) ; x R ( A 1 / 2 ) ¯ , A 1 / 2 x R ( A 1 / 2 ) = 1 .
Now, let λ = ( λ 1 , , λ d ) Ω ( T ˜ ) . Then, there exists x R ( A 1 / 2 ) ¯ satisfying
A 1 / 2 x R ( A 1 / 2 ) = 1 and λ i = T i ˜ A 1 / 2 x R ( A 1 / 2 ) , i { 1 , , d } .
Since the subspace R ( A ) is dense in R ( A 1 / 2 ) , then there exists a sequence { ξ n } , which may be assumed to be in R ( A 1 / 2 ) ¯ (because of the fact that H = N ( A ) R ( A 1 / 2 ) ¯ ) such that A 1 / 2 x = lim n + A ξ n . Thus, by taking (44) into consideration, it holds that
lim n + A ξ n R ( A 1 / 2 ) = 1 and λ i = lim n + T i ˜ A ξ n R ( A 1 / 2 ) ,
for all i { 1 , , d } . Now, set θ n : = ξ n A ξ n R ( A 1 / 2 ) . Clearly, we have A θ n R ( A 1 / 2 ) = 1 . Further, by using (45), it can be checked that
lim n + T i ˜ A θ n R ( A 1 / 2 ) = λ i , i { 1 , , d } .
This implies that, through (42), λ Ω A ( T ) ¯ and, therefore,
Ω ( T ˜ ) Ω A ( T ) ¯ .
From (43) and (46), we deduce that Ω ( T ˜ ) ¯ = Ω A ( T ) ¯ . Therefore, we infer that
T A = T ˜ .
On the other hand, since T = ( T 1 , , T d ) B A ( H ) d is an A-doubly-commuting d-tuple of A-hyponormal operators, then
T i T j = T j T i , i , j { 1 , , d } , T i A T j = T j T i A , 1 i j d .
and T k A T k A T k T k A , k { 1 , , d } .
Therefore, by applying Proposition 3 together with Lemma 3 (ii), we have
T i ˜ T j ˜ = T j ˜ T i ˜ , i , j { 1 , , d } and ( T i ˜ ) * T j ˜ = T j ˜ ( T i ˜ ) * , 1 i j d .
and ( T k ˜ ) * T k ˜ T k ˜ ( T k ˜ ) * , k { 1 , , d } .
Hence, T ˜ = ( T 1 ˜ , , T d ˜ ) is a d-tuple of doubly-commuting hyponormal operators on the Hilbert space R ( A 1 / 2 ) . Therefore, by (13), we have
T ˜ = T ˜ e .
This completes the proof by taking (41) and (47) into consideration. □
Remark 7. 
Note that the converse of Theorem 3 need not be correct as shown in the next example.
Example 1. 
Let us consider the same matrices in M 3 given in Remark 1, i.e., A = 1 0 0 0 2 0 0 0 1 , T 1 = 0 1 0 0 0 0 0 0 0 and T 2 = 0 0 1 0 0 0 0 0 0 . By Remark 1, we have ( T 1 , T 2 ) A = 1 . Further, we see that
( T 1 , T 2 ) e , A = sup ( λ 1 , λ 2 ) B 2 λ 1 T 1 + λ 2 T 2 A = sup ( λ 1 , λ 2 ) B ¯ 2 λ 1 T 1 + λ 2 T 2 A ,
where B ¯ 2 means the closed unit ball of C 2 . So, by using (4) and making direct calculations, we show that
( T 1 , T 2 ) e , A = sup | λ 1 | 2 + | λ 1 | 2 1 sup | x | 2 + | y | 2 + | z | 2 = 1 λ 1 2 y + λ 2 z .
By making use of the Cauchy–Schwarz inequality, it can be easily checked that ( T 1 , T 2 ) e , A 1 . On the other hand, by using (48) and then (4), we see that
( T 1 , T 2 ) e , A T 2 A = 1 ,
from which ( T 1 , T 2 ) e , A = 1 . So,
( T 1 , T 2 ) A = ( T 1 , T 2 ) e , A = 1 .
However, it can be verified that T 1 A T 2 T 2 T 1 A and this ( T 1 , T 2 ) is not an A-doubly-commuting 2-tuple of A-hyponormal operators.
Remark 8. 
According to our proof in Theorem 3, we remark that the equality
( T 1 , , T d ) A = ( T 1 ˜ , , T d ˜ ) ,
holds for every d-tuple of operators ( T 1 , , T d ) B A 1 / 2 ( H ) d . Here, T k ˜ B ( R ( A 1 / 2 ) ) and verify Z A T k = T k ˜ Z A for all k { 1 , , d } . Note that (49) provides an improvement of a result of the second author in [19] since B A ( H ) is in general a proper subspace of B A 1 / 2 ( H ) .
In order to derive an important consequence from Theorem 3, we first introduce the following definition that is inspired by the work of G. Popescu [16].
Definition 5. 
For T = ( T 1 , , T d ) B A 1 / 2 ( H ) d , we define a new A-joint numerical radius and a new A-joint spectral radius of T by setting
r e , A ( T ) = sup ( λ 1 , , λ d ) B d r A ( λ 1 T 1 + + λ d T d ) ,
and
ω e , A ( T ) = sup ( λ 1 , , λ d ) B d ω A ( λ 1 T 1 + + λ d T d ) .
Remark 9. 
If A = I , then r e , I ( · ) will simply be denoted by r e ( · ) . Further, it is worth mentioning that the equality
r ( T ) = r e ( T )
holds for every commuting operator tuple T = ( T 1 , , T d ) B ( H ) d (see Theorem 2.1 in [29] or [2]).
Now, as an application of Theorem 3, we state the following result.
Theorem 4. 
Let T = ( T 1 , , T d ) B A ( H ) d be an A-doubly-commuting d-tuple of A-hyponormal operators. Then
r A ( T ) = T A = ω A ( T ) .
Proof. 
Since T = ( T 1 , , T d ) B A ( H ) d is a d-tuple of A-doubly-commuting A-hyponormal operators, then in particular we have
T k A T l = T l T k A , 1 k l d and T m A T m A T m T m A ,
for every m { 1 , , d } . Now, for λ = ( λ 1 , , λ d ) C d , we let S λ : = m = 1 d λ m T m . Clearly, S λ B A ( H ) . By making simple calculations and using (52), we see that
S λ A S λ = i = 1 d j = 1 d λ j λ i ¯ T i A T j = i = 1 d | λ i | 2 T i A T i + i = 1 d j = 1 , j i d λ j λ i ¯ T i A T j A i = 1 d | λ i | 2 T i T i A + i = 1 d j = 1 , j i d λ j λ i ¯ T j T i A = S λ S λ A .
Hence, S λ is an A-hyponormal operator. This implies that, by (7),
r A ( S λ ) = ω A ( S λ ) = S λ A ,
for all λ = ( λ 1 , , λ d ) C d . If we take the supremum over all λ B d in the last equalities, then we obtain
r e , A ( T ) = ω e , A ( T ) = T e , A .
Taking into account relation (25), we have
ω A ( T ) = sup x H , x A = 1 j = 1 d | T j x , x A | 2 = sup x H , x A = 1 sup ( λ 1 , , λ d ) B d j = 1 d λ j T j x , x A .
This implies that
ω A ( T ) = sup ( λ 1 , , λ d ) B d sup x H , x A = 1 j = 1 d λ j T j x , x A = sup ( λ 1 , , λ d ) B d ω A j = 1 d λ j T j .
This proves
ω A ( T ) = ω e , A ( T ) .
In view of Theorem 3, we have T e , A = T A . So, all that remains to be proven is that r A ( T ) = r e , A ( T ) . Since we have T = ( T 1 , , T d ) B A ( H ) d B A 1 / 2 ( H ) d , then by applying Proposition 3 we deduce that for each j { 1 , , d } there exists T j ˜ B ( R ( A 1 / 2 ) ) such that Z A T j = T j ˜ Z A . Hence, another application of Proposition 3 shows that
r e , A ( T ) = sup ( λ 1 , , λ d ) B d r A λ 1 T 1 + + λ d T d = sup ( λ 1 , , λ d ) B d r λ 1 T 1 + + λ d T d ˜ = sup ( λ 1 , , λ d ) B d r λ 1 T 1 ˜ + + λ d T d ˜ ,
and so,
r e , A ( T ) = r e ( T ˜ ) , where T ˜ = ( T 1 ˜ , , T d ˜ ) .
Since T is an A-doubly-commuting operator tuple, then it is commuting. So, similar to the proof of Theorem 3, we find that T ˜ is a commuting d-tuple of operators in the Hilbert space R ( A 1 / 2 ) . Therefore, by (51), we conclude that r e ( T ˜ ) = r ( T ˜ ) . Further, by Proposition 3 (4), we have r A ( T ) = r ( T ˜ ) . Hence, by taking (55) into account, we deduce that
r A ( T ) = r e , A ( T ) ,
as desired. Thus, combining (53) with (54), (56) and (40) yields the desired result and the proof is complete. □

5. Conclusions

In this paper, we introduced a definition that is a generalization of (12). For T = ( T 1 , , T d ) B A 1 / 2 ( H ) d , the A-Euclidean seminorm of T is given by
T e , A : = sup ( λ 1 , , λ d ) B d λ 1 T 1 + + λ d T d A .
Consequently, our objective was to study a new joint norm of tuples of operators which generalizes the joint norm given in (12) and define the class of doubly-commuting tuples of hyponormal operators acting on an A-weighted Hilbert space, where A is a positive operator that is not assumed to be invertible. The motivation for our investigation comes from the recent paper [17].
This article was structured as follows: In Section 3, we investigated a new joint seminorm for d-tuples of A-bounded operators. An alternative and easy proof of a well-known result due to G. Popescu [16] was established. In Section 4, we give several characterizations related to the operators from B A 1 / 2 ( H ) and the operators from B ( R ( A 1 / 2 ) ) . For the A-doubly-commuting d-tuple of hyponormal operators, we proved the equalities T e , A = T A and r A ( T ) = T A = ω A ( T ) .
In this paper, the ideas and methodologies used may serve as a starting point for future studies in this field. We will look for other connections of these seminorms for d-tuples of A-bounded operators by studying other possible characterizations. In future work, we will generalize the results given a countable collection of operators.

Author Contributions

The work presented here was carried out in collaboration between all authors. All authors contributed equally and significantly to the article. All authors contributed to the manuscript. All authors have read and agreed to the published version of the manuscript.

Funding

The first author extends her appreciation to the Distinguished Scientist Fellowship Program at King Saud University, Riyadh, Saudi Arabia, for funding this work through Researchers Supporting Project number (RSP2023R187).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Acknowledgments

The authors express their deepest gratitude to the anonymous reviewers for their important comments and suggestions, which significantly improved our article.

Conflicts of Interest

The authors declare no conflict of interest.

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MDPI and ACS Style

Altwaijry, N.; Feki, K.; Minculete, N. A New Seminorm for d-Tuples of A-Bounded Operators and Their Applications. Mathematics 2023, 11, 685. https://doi.org/10.3390/math11030685

AMA Style

Altwaijry N, Feki K, Minculete N. A New Seminorm for d-Tuples of A-Bounded Operators and Their Applications. Mathematics. 2023; 11(3):685. https://doi.org/10.3390/math11030685

Chicago/Turabian Style

Altwaijry, Najla, Kais Feki, and Nicuşor Minculete. 2023. "A New Seminorm for d-Tuples of A-Bounded Operators and Their Applications" Mathematics 11, no. 3: 685. https://doi.org/10.3390/math11030685

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