Lattice Sequence Spaces and Summing Mappings

1.    Introduction and Background

The spaces of sequences with values in the Banach lattice are intimately related to the summability of operators between Banach lattices. For example, the positive ( p, q ) -summing operators, introduced by Blasco [1], are the continuous operators which take positive weakly p -summable sequences l _p ,m( X ) into q -summable sequences l q ( E) (see also [2]). In [3], Achour-Belacel introduced the notion of positive strongly ( p, q ) -summing operators to characterize those operators whose adjoints are positive ( q * ,p * ) -summing operators.

In [4] and [5] the authors defined the space of positive strongly p -summable sequences lp ( X ) (initially introduced by Cohen for the Banach spaces [6]), as well as the space of positive unconditionally p -summable sequences l u | ^ | ( X ) •

(0 2025 Dahmane, A. and Toufik, T.

Tensor products have proved to be a useful tool for the theory of operator ideals. Indeed, the excellent monograph [7] deals with the theory of the tensor product point of view and provides many applications to the study of the structure of several spaces of summing linear operators.

The following characterizations provide nice examples of how tensor products come into the theory of summing operators:

• •    An operator T : X ^ Y is absolutely p -summing (see [8]) if and only if 1 0 T : I p 0 _£ X ^ I p 0 Д _Р Y is continuous, where A _p satisfies A _p ( ^ П ₌₁ e i 0 X i ) = ( 53 n=i ||x i|| ^p ) ^{1 /p} and £ is the injective tensor norm (see [7]).

• •    Let 1 < p <  от . An operator T : X ^ Y is the Cohen p -nuclear ( p -dominated) if and only if 1 0 T : l _p 0 _£ X ^ l _p 0 _n Y is continuous and n is the projective norm (see [6] and [7]).

• •    An operator T : X ^ Y is strongly p -summing if and only if 1 0 T : l p 0 & _p X ^ I p 0 _n Y is continuous (see [9]).

The interplay between tensor products and positive summing operators have not been explored yet. In this paper, first we utilize sequences in Banach lattice spaces to define and characterize certain classes of positive summing operators. Then we describe these classes in terms of the continuity of the canonically defined tensor product operator 1 0 T : l p 0 _a X ^ l _p 0 e Y for adequate p and tensor norms a and в .

Our results are presented as follows. After this introductory section, Section 2 is devoted to providing new properties of the positive ( p, q ) -summing operators. Particularly, we prove that these operators are continuous operators transforming positive lattice unconditionally p -summable sequences l u | ^ | ( X ) into q -summable sequences l q ( E). In Section 3, utilizing the Banach lattice of positive strongly p -summable sequences, we present a novel characterization of positive strongly ( p, q ) -summing operators. Furthermore, we demonstrate that this class is equivalent to the class of ( p, q ) -majorizing operators introduced in [10]. In Section 4, we study the notion of the positive Cohen ( p, q ) -nuclear operators. We explore the summability properties of these operators by defining their corresponding operators between spaces of positive weakly p -summable sequences l _p, | _W | ( X ) and strongly positive strongly p -summable sequences l^ ( Y ) . In the final section, we describe these classes in terms of the continuity of an associated tensor operator that is defined between tensor products of sequences spaces.

We use standard notation for the Banach lattices (see [11, 12]). If X is an ordered set, the usual order on X N * is defined by x = ( x n ) _{n e} N * ^ 0 О x n ^ 0 for each n G N * . Recall that the Banach lattice X is an ordered vector space equipped with a lattice structure and the Banach space norm satisfying the following conditions: ||x || C |y| whenever |x| C |y| for all x,y G X , where |x| = x V ( —x ) . Note that this implies obviously that for any x G X the elements x and |x| have the same norm. We denote by X ₊ = {x G X, x ^ 0 } . An element x of X is positive if x G X ₊ . For x G X let x + := x V 0 , x - := ( —x ) V 0 be the positive part and the negative part of x , respectively. For any x G X , we have the following properties x = x + — x - and |x| = x + + x ^- .

The dual X ^∗ of a Banach lattice X is a complete Banach lattice endowed with the natural order

x* C x2 ^^ (x*,x) C (x2>x) for all x G X+, where {•, •) denotes the bracket of duality.

By a sublattice of a Banach lattice X we mean a linear subspace A of X so that sup {x,y} = x V y belongs to A whenever x,y G A. The canonical embedding iE : X ^ X** such that {iE (x) , x*) = (x*,x) of X into its second dual X** is an order isometry from X into a sublattice of X∗∗, see [11, Proposition 1.a.2]. If we consider X as a sublattice of X∗∗ we have for xi,X2 € X xi C x2 ^^ (x1,x*) C (x2,x*) for all x* € X+.

Throughout this paper X and Y are Banach lattices, E and F are Banach spaces. We say h : X ^ Y is a vector lattice homomorphism if it is a linear operator such that h ( x i V X 2 ) = h ( x i ) V h ( x 2 ) for all x i ,x 2 € X . An one-to-one, surjective vector lattice homomorphism is called vector lattice isomorphism. A linear operator S : X ^ Y is called positive, if S ( x ) ^ 0 for any x ^ 0 . A map T : X ₊ ^ Y ₊ is called additive if for every x,y € X . we have T ( x + y ) = T ( x ) + T ( y ) . We call T positively homogeneous if for each A € R . and every x € X + , we have T ( Ax ) = AT ( x ) . The space of all bounded linear operators from E to F is denoted by L ( E,F ) and it is the Banach space with the usual supremum norm. The continuous dual space L ( E, K ) of E is denoted by E * , whereas B e denotes the closed unit ball of E. The symbol E = F means that E and F are isometrically isomorphic.

Let 1 C p C ro, we write p * the conjugate index of p , that is 1 /p +1 /p * = 1 . As usual l _p ( E) denotes the vector space of all absolutely p -summable sequences, with the usual norm ||-|| p and 1 _р , _ш ( E) the space of all weakly p -summable sequences with the norm ||( x _n ) _{n e} N * | _рш = sup x . g B E , | ( (x _n , x * ) ) _{n e} N * ||p . The closure in 1 _р,_ш ( E) of the set of all sequences in E which have only a finite number of non-zero terms, is a Banach space with respect to the norm IHlpw . We denote this space by l ^u ( E ) . Let c o be the space of scalar sequences ( A n ) _{n e} N * , such that A n —> 0 . The space

(со)ш (E) := {(xn)neN* C E : ((xn,e»neN* € co V£ € E*J is a closed subspace of lM,^ (E) = lM (E) (see [13, §19.4]). It is well known that 1р,ш (E) is an isometrically isomorphic to L(lp* ,E) for 1

IK xnk e N * 1 _Р« , _И ^C1

M

x ^* _n ( x n )

n=1

M sup      ^|xjn(xn)| .

ll ( ^x n ) n e N * ^.ш ^C1 n = 1

The following fact, discussed in [14], is well-known lp,u(E*) = [lp (E)]* and [lp* (E*)] = [1р,ш(E)]* .

(1.1)

(1.2)

Moreover, it is well-known that

[ l _p ( E )] * = l p * ( E* ) for 1 C p< ro and [ c o ( E )] * = l 1 ( E* ) .

Sequences in Banach lattice spaces. Consider the case where E is replaced by a Banach lattice X. The space of positive weakly p-summable sequences was introduced in [4] by lp,H(X) = {(xnUN € XN* : f(x*, |xn|)p < ro Vx* € X^ , endowed with the norm

/ ^M         \p

||( ^x n ) n e N * lip, |_Ш| =   ^suP      £^{( x} * , ^|x n ^{|) p} .

x * e Bx ; V“i         J

Also, ( с о ) | _ш | ( X ) is a closed vector lattice subspace of i _Mj _w | ( X ) . Then ( l _p, | _w | ( X ) , || • ||p, | w | ) is a Banach lattice (see [4, 5]). Moreover, we have ( I o ) ||( x n ) n _e N * H _pw < | ( x n ) n e N * H _p | ^ | for all ( x n ) n e N * e i p_M ( X ) .

( I 1 ) If ( x n ) n _e N * >  0 , we have

(1.3)

IKx nW lip ,M = l ( x n Un* lip , _ш

We define lp,H(x*) = {(xnw e (X*)n*

M

^ (x, UDp < ™vx e x ₊ n=1

and

HUUn* Um = sup x ^ B x +

( M \ P

(E wxH

(1.4)

Then l _Pj | ^ | ( X * ) with this norm is a Banach lattice (see [4, 5]).

Let lu|w|(X) denote the closed sublattice of ipjw|(X) defined by lU,M(X) = {(xnUn* e XN* : linm |(xkUnU^M = 0} •

In this case we say that (xn)neN* is positive unconditionally p-summable sequences. Let in (X) = {uUn* e Xn*

M

E n=1

l(x n , |x n |)| < » v ( x n ) n e N * e ( i _p ._)M ( X * )) ⁺

and

M

IIUUn* ^H i ⁿ (x ) =       sup        E«’ Ы> '             (1-5)

⁽ x n ) n e N * ^{e B} [ i _p* , | „ | ( x * )]+ n=1

In this case we say that ( x _n ) _{n e} N * is positive strongly p -summable sequences. Then i^ ( X ) with this norm is a Banach lattice [5]. For convenience let us denote i 1 (X) = 1 П ( X ) = i i ( X ) .

By (I o ) and (I 1 ), wehave ( I 0 ) H ( X n ) n e N * H^ X) < H ( X n ) n e N * H _{l p} ( _X > for all ( X n ) n e N * e i p (X) .

( I 1 ) If ( x n ) n _e N * >  0 , then ( x n ) n _e N * e ip ( X ) if and only if ( x n ) n _e N * e i p (X) , and

(1.6)

IIUUn* ^H l ⁿ (X) = ^{H ( x} n ) n e N * lll _p ( X > •

Moreover, we have the following results due to [4, 15].

Proposition 1.1 [15, Proposition 3.1 and Proposition 3.2] . Let X be a Banach lattice and 1 < p <  to . Then

^{( a )}

i p ( X * ) = {(x n ) n e N * e ( X *) ⁿ *

M

^: ^ ^|(x n , ^|x nD | <  ^to, ^{v ( x} n ) n G N * ^{e ( i} p * , | u | ( ^X )) n=1

and for each ( x n ) _{n e} N * e i p ( X * ) ,

M

||⁽x n ⁾ n e N * ^H i ⁿ ( x * ) =          sup           £⁽ x n , |x n |).

^{( X n )} n e N * ' B .p \ ■ n=1

( b )

i ; ( X * ) = {^Un* G ( X * ) ^N * : f |(x n , |x n i>l < от V ( x n ) n e N * G (^( X )) ⁺ }

and for each (x^ n e N * G Ip ( X * ) ,

IKx nke N * ^ i ; (X * ) =         ^suP

^{( x} n ) n e N * ^{e B} [ lu p ^∗ ,ω

M

£ (x n |x n |>.

( X )]+ n=1

Lemma 1.2 [4] . Let X be a vector lattice, ( Y, C ) be an ordered vector space, such that Y = C - C and T : X ₊ —> C be a positive, homogeneous and additive bijection. Then Y is a lattice space and T can be uniquely extended to a lattice isomorphism from X onto Y .

Theorem 1.3. Let X be a Banach lattice and 1 < p <  от.

• (i)    The Banach lattice l _p * , | _w | ( X * ) is lattice and isometrically isomorphic to ^B( X )] * .

• (ii)    The Banach lattice [ip * (X * ) ] is lattice and isometrically isomorphic to [l u | _w | ( X )] [15, Corollary 3.3 and Corollary 3.4] .

<1 (i) Let 1

, we define the mapping

T : I p * _M(X * ) -4 [i; ( X )] * ,  x * = ( x n ) n e N * -^ T ( x * ) = T x * ,

where T _x ∗ is the linear functional defined by

M

T x * ^{: l}p ^{( X} )     >  K    ^{( x} n ) n e N * ¹ ^ T x * ^{(( x} n ) n e N * ) =      x ^* n ( x n ) .

n=1

The map T is clearly a positive map from (i _p * , | _w | ( X * )) ₊ to [i j ( X )] ^ , and it is homogeneous, additive and injective. To see that it is surjective, note that if S G [l j ( X )] ^ and

I n : X -^ i p ( X >,   x -^ (0 ,...,x, 0 ,... ) ,

х П = S о I n G X ++ for all n G N * .

Then

СЮ

M

T (S o I n ) n e N * (( x n ) n e N * ) = £(S о In) ( x n ) = £ S ( I n ( x n )) = S ( I 1 ( x i )) + • •• + S ( I n ( x n ) + •••

n=1

n=1

= ^{S (( x} 1 , ⁰ ;.-- ⁾) + • • • + ^{S ((0 ,} . . . , ^x n , • • • ⁾⁾ + • • • = ^{S (( x} n ) n e N * ) •

For x G B X , we get

(M           \ p*    / M

E (xn ,x>ip‘   = Ei® о w(xr n=1n=1

sup

^{( a} n ) n e N * ^e B p

M

^ a n S о r n ( x)

n=1

sup      |S (( a n x ) n e N * ) | <  l|S II

(an)neN* e Bp sup     |(anx)neN* hip(X)-

^{( a} n ) n e N * ^e B + p

We need to estimate the latter expression. Note that га ll(anx)neN* ||ip(x) =          sup          ^(xn lanxn|)

(x n ) n e N * ^{e B} [ l _p* , | „ | ( x * )]+ n=1

га

=         sup          ^|a n |n, |xn|)

^{( x} n ) n e N * ^{e B} [ i _p* , | „ | ( x * )]⁺ n=1

C H( ^a n ) n e N * lip            ^sup            ll( x n ) n e N * llp * ,M ^ ||( ^a n ) n G N * ||p -

⁽ x ^{n )} n e N * ^{e B} [ lp * ,_H ( x * )l ⁺

Then

i

( ^ra                 p

£ |S ◦ I n )( x ) | ^p *       C IISII      sup

l ( a n Un* ||p = ||S||.

\n =1               /            ^{( a} n ) n e N * ^ B t p

Hence, by (1.4), ( x n ) n _e N * G ( i _p* , | _w | ( X * )) ₊ .

Since l _p*, | _Ш | ( Х * ) is the Riesz space and [ i p" ( X ) ] * = [ i p" ( X ) ] + — [ i p" ( X ) ] + , it follows from Lemma 1.2 that [l ⁿ ( X ) ] * is a lattice space and that T is a lattice isomorphism from l _p*, | ^ | ( X * ) onto [1 П ( X ) ] * .

For ( x n ) n e N * G l p., | ^ | ( X * ) , we have

I T (( x n ) n G N * ) I [ l n (X) ] * = IT (KW ) l L (i j (X ), K ) = I|T (( x n ) n G N * ) 11 [ i n ( x ) ] * = I T ( | ( x n ) n e N * | ) | [ 1 П (Х) ] * = | ( |x n | ) n G N * ||p*,_H = | ( x n ) n G N * | p*,_H -

This means that T is an isometry from l _p*, | _w | ( X * ) onto [ip"( X )] * . >

2.    Positive (p, q)-Summing Operators Generated by iu ^(X)

Let 1 C q C p <  то- Following [1, Definition 1] and [3, Proposition 3.2 (1)], an operator T : X —> F is said to be positive ( p, q ) -summing if there exists a constant C >  0 , such that for every x i ,..., x _n G X, we have

(2.1)

| ( T ( X i )) n=i | p C C | ( X i ) n=1 | q,_H -

For q < p = то, we get sup |T(Xi)| C C |(Xi)n=1|q,H-ICiCn

We shall denote by A _p,_q ( X, F ) the space of positive ( p, q ) -summing operators. This space becomes a Banach space with the norm | • ||д given by the infimum of the constants verifying (2.1). For p = то and 1 C q < то we consider Л _{га q} ( X, Y ) = L ( X, F ) and |T IKq = |T||-

Now, we give characterizations of these classes in terms of transformations of lattice vectorvalued sequences.

Proposition 2.1 [1, Proposition 2] . Let T : X —> F be an operator and 1 C q C p C то . The following properties are equivalent.

• (1)    T G Л p,q ( X,F ) .

• (2)    The associated operator T : i _q, | _w | ( X ) —> i _p ( F ) given by T (( x i ) i e N * ) = ( T ( x i )) i e N * , ( x i ) i e N * G i _q, | _w | ( X ) is well-defined and continuous.

In this case ^{|T |} Л p,q = ||^T ||-

Theorem 2.2. For a continuous linear T G L ( X, F ) and 1 C q C p C от , the following conditions are equivalent.

• (i)    T G A p,q ( X,F ) .

• (ii)    The sequence ( T ( x i )) , _e N * G l p ( F ) whenever ( x , ) , _e N * G ^^( Х ) •

• (iii)    The induced map

T : Ы.Х) ^ lp (F), T ((xiW ) = (T (Xi)W, is a well-defined continuous linear operator and ||T||д = ||T||-

О (i) ^ (ii) Let x = ( x , ) , _e N * G l^ C X ) . We have

l(T(Xi))n=1lp C C |(Xi)n=1|q,H, for all n G N*. So, if mi > m2, then kt (x.))m1 - (t мед=|(t ex, )e,„2+iiip c c i(x,E„tiiq,H.

We conclude that ( y n ) n e N * with y _n = ( T ( x , )) П=1 is the Cauchy sequence in l _p ( F ) and so converges to some ( z , ) i e N * ^{G l} p ^{( F} ) ■

Given e >  0 , we can find N q G N * so that n ^ N o ^ | ( T ( x i )) П ₌₁ — ( z i ) , _e N * | _p <  e. So, for a fixed i o G N * , we have |T ( x , ₀ ) — z , 0 1| <  e. We conclude that T ( x , 0 ) = z , 0 ■ Hence ( T ( x i )) i _eN * = ( z i ) i _e N * G l p ( F ) ■

• (ii)    ^ (iii) Is it clear that T is linear implies that T is linear, for show that T is continuous we showing that T has a closed graph. Suppose that the sequence ( T ( x , )) , _e N * G l _p ( F ) whenever ( x , ) , _e N * G lu^^X ) and let ( ( x k ,T ( x k )) ) k _eN * be a convergent sequence in the Cartesian product l u^ | ( X ) x l _p ( F ) that is, (( x k ) _{i e} N * ,T (( x k ) _{i e} N * ) — > ( x,y ) . So

^x k ---^ ^x = ^{( x} n ) n e N * ^{G l} u, | _Ш | ( ^Х ) >                                ⁽²-²⁾

T ( x k )    ■ y = ( y nU N G l p ( F ) ■                          (2.3)

From (2.2), for all e >  0 there exist k o >  0 , such that

|x * ( x k — x i ) | ^q C ^ |x * || ( x k — x , ) | ) C ^ ( x * | ( x k — x , ) | ) i=1

C sup x∗∈B ∗ X+

∞

£ (^x*^{| ( x} k ^— x i ) ¹)⁷

c ip—

whenever k > ko, x* G Bx+ and for all i G N*. In this way, by the Hanh-Banach theorem, we get

|xk — xi||^q= sup |x*(xk — xi)|^qC 2^qsup |x*(xk — xi)|^qC 2^qEq,        (2.4)

x∗∈BX∗                       x∗∈BX∗ whenever k > ko and for all i G N*, then we have xk —> x, G X for all k —> от. How T is continuous, we find limT(xk) = T(x,) for all i G N*.

From (2.3), for all e > 0 there exist kg > 0, such that

∞

||t(x^k) - yi| C E ||t(xk) - yi|| C |(T(xk»*"* - (yiMl = IM) - уЦ C ep, i=1

whenever k > kg and for all i G N*, we find lim T(xk ) = yi for all i G N*.

(2.5)

k

From (2.4), (2.5) and uniqueness of the limit, it follows that

T (x) = (T (xi))i_eN* = (yi)ieN* = y.

This implies that the linear mapping T has a closed graph.

• (iii)    ^ (i) is straightforward. >

3.    Positive Strongly (p, q)-Summing Operators Generated by ln(Y)

Let 1 C q C p C №. Recall that an operator T G L(E,Y) is called positive strongly (p, q)-summing [3, Remark 4.2] if there exists a constant C > 0, such that for all finite sets, (xi)n₌₁C E and (y*)n₌₁C (Y*) ⁺, we have

n

E l(T(x.),y*)| C C|(x)n=i|q|(у'Ь1|,.,.                    (3.1)

i=1

We shall denote by D+_q(E,Y) the space of positive strongly (p, q)-summing operators or D+(E,Y) if p = q, the space of positive strongly p-summing operators. This space becomes a Banach space with the norm | • |^+ given by the infimum of the constants verifying (3.1).

Lemma 3.1. The operator T G Dpq(E,Y) if and only if there exists a constant C > 0, such that for all finite sets, (xi)n=₁C E and (у*)П=1 C (Y*)+, we have

• l(T(Xi)n=1|in(Y) C C|(Xi)n=1|q.                                (3.2)

• <1 Let T G Dpq(E,Y), then there exists a constant C > 0, such that for all finite sets, (Xi)n=1 C E and (y*)n=1 C (Y*) ⁺, we have

n

El(XiЬ.*>| C Cl(x)n=1lql(y*)i=1lp-,-.

i=1

For each z G Y and z* G Y*,

• |z*|(|z|)=sup{|g*(z)| : |g*| C |z*|} .                                 (3.3)

Now let (y*)n=1 G (l_p*,|w|(Y*))+ and let e > 0. From (3.3), there exists, for each 1 C i C n, an element g* G Y*, such that |g*| C |y*|, and

|y*|(|T(xi)|) C |g*(T(xi))| + -.

n

Note that (g*)n=1 e (^(Y*))+. Then nn  n

E (i t (xi )i ,y*) = E< । t (x->। , ।y*i i(xi),gi i| +' i=1                 i=1                   i=1

• < C [|(Xi ),= 1|q |(g* )n=1llp^ ]+ £ C C [|(Xi ),= 1|q 11^=111^ ]+ ^

Then

|(T (_Xi)n=1lln₍Y) C C |(Xi),= 1|q + £.

Conversely, directly by |(T(хДу*)| < (|T(xi)|,y*) for every i. >

As in classical cases, the natural approach to presenting the summability properties of positive strongly (p, q)-summing operators by defining the corresponding operator between appropriate lattice sequence spaces.

Proposition 3.2. Let T : E —>Y be an operator and 1 C q C p C от. The following properties are equivalent:

• (1)    T e D+q(E,Y).

• (2)    The associated operator T : lq(E) —>IJ(Y) given by T ((xi)_ieN*) = (T(xi))_ieN*, (xi)ieN* e lq(E) is well-defined and continuous.

In this case ^|T||D,+_q= ^lT||.

О For the necessity, let T e D+ (E,Y), (xi)ieN* e lq(E) and (y*)ieN* € (lp*,n(Y*))⁺-Then by Lemma 3.1, we have

∞n

C |TID+q|(xiW lq|(у*)геж lp.^, which implies

∞ sup        £ (^|T(xi)^*) ^{C lT} hp+q IK^Xi)ieN* ^|q .

(yOieN* ^eB[ip*,_k|(Y*] ⁺i=1

Consequently, we obtain

∞

I(T (Xi ))i₆N. |_f, (Y) =        sup E<|T (Xi)U’I « IT |D+, IltxiW lq ,

⁽yi^)ieN* ^eB[ip*,|„|(Y *]+ i=1

and therefore T is continuous with norm C |T|^+ .

In order to prove sufficiency, suppose T is well-defined and continuous and assume that T / Dp+q(E,Y). Then for each n e N*, we may choose a finite sequence (xi,,)"^ C E, such that ||(xi,n)mn1||q C 1 and |(T(xi,n))mn1 |in(y) > 2n, which implies mn

E<|T(Xi..J|,C,I > 22n

(3.4)

i=1

for some (y*,)"^ e (lp*,|w|(Y*))+, such that |(y*n)mn1|p*iW C 1. Let (zj)jeN* be the sequence x^m1 ,m m2 ,..JMmn ,.J

2¹Л=1 Ч 2²Л=1 ’   4 2ⁿЛ=1 ’   )

(^x1,1 ^x2,1

Xmi,1 X1,2 X2,2

^xm2,2

X1,n X2,n 2n ’ 2ⁿ

xmn,n

2ⁿ

,... .

We have

∞ ^mj

EE || > || j=1i=1

1 ^{q q}

1                   1

∞                 ^q∞ ^q

E j i(x.jI < (e 21s I < i- j=1                         j=1

Then, (zj)j_gn* G lq(E). However, T((zj)j_GN*) / Ip(Y). In order to see this, consider the sequences

((y*i\^mi(yi*2\^m2     (yinVn

Oft ^)jGN' -^21 Х=, A 22 C1 ’-A 2'4=1 ■•••/

Clearly ⁽^j )jGN* G B(ip* ,_H(Y *))+ . Then

^{∞ mj}    y ^∗

IK^j )jGN* ^Hp.,_M= ySB^pIjT    (x, j

1 p∗ p∗

∞     ^mj          ^p∗     ∞

E j* E<xIp* I < (E j*

j=1 ²   i=1                    j=1 ²

p^∗

< 1.

By (3.4) it turns out that

IIT ((zj jGN* ^)Hiⁿ(Y) = ^H(T(zj Dj GN* Hi;- (Y) = SUP ⁽'j)jeN* ^H

p^∗,ω

∞

E (|t (zj Uji

<1 1=1

_{∞            ∞}    mj

>E(|t (zj )l,eI = E — E«T (xij )Kj I = ”' 2

j=1

j=1     i=1

which according (1.5) is a contradiction with the fact that T maps lq(X) (continuously) into lp(Y). Since

----

.—-

H(T (Xi ))iGN* ) Ilin (Y) = l|T ((Xi)iGN* )llln (Y ) < IITII ll(Xi W llq ,

we have HTHD+q < HTH- >

In the following result, we characterize the class of positive summing linear operators and positive strongly summing linear operators by utilizing the adjoint operator. For the proof of this result, we will utilize the duality of lattice sequence spaces. Theorem 3.3 was established in [3, Theorem 4.6], and the proof provided there is direct. Using Theorem 1.3, the formula (1.2), Proposition 3.2 and taking into account that the adjoint of the T : lq(E) —>lp(Y) can be

-—-

-—-

identified with the operator T* : lp*j^(Y*) —>lq*(E*); T*((y*)ieN*) = (T*(yi*))iGN*, provide an alternative proof of the results in Theorem 3.3.

we

Theorem 3.3. Let T : E —>Y be an operator and 1 < q< p<

• (1)    The operator T belongs to A_p,_q(X,F) if and only if its D+p* (F *,X *). Furthermore, \\T K, = IT *HD+ ,p*.

• (2)    The operator T belongs to D+_q(E,Y) if and only if its

TO.

adjoint T ^∗belongs

adjoint T ^∗belongs

to

to

Aq*,p*(Y*,E*). Furthermore, HTHD+q = ||T*HЛq*,p.

∗^.

<1 (1) Let T E L(X,F) and T* E L(F*,X*) its adjoint. Suppose that T E Ap,q(X,F), then by Theorem 2.2 T : lu^X) —>l_p(F) is continuous with ||T||д_р9= ||T||. By (1.2) and Theorem 1.3, the following diagram commutes

∗ lp (F)*

J1 t

U (F*)

U  l^ui |(X)^:

q,|ω| t J2

∗

U  l^ (X*)

i.e., T* о J1 = J₂ о T*, where J1 is an isometric isomorphism and J₂is an isometric lattice isomorphism, such that Ji((zn)n_eN*)((zn)n_eN*) = f ((zn)n_eN*) = £“₌₁{zn,zn), i = 1, 2, with the inverse Ii, defined by Ii (f) = (f о I_n)_neN* = (zn)_neN*. In fact, the map T*, defined by ^(yn)neN* ^ ^(T* ⁽yn))neN* , ^{let (}yn)neN* ^{E l}p* ^(F*), ^{then for} all ^(xn)neN* ^{E l}U,m^(X)i

(T* о Ji)((yn)_neN*) ((xn)neN*) = Ji((yn)neN*) T ((xn)n_eN*))

∞∞

= J1((yn Un* ) ((T (Xn))neN* ) £ {уП ,T (Xn)) £ {T *(уП ),xn) = J2((T*(уП )W ) ((Xn)neN* ) n=1           n=1

= J2(T* ((yn)neN* )) ((Xn)neN* ) = (J2 о T*) ((yn)neN* ) ((Xn)neN* ) ,

• i .e., T* о J1 = J2 о T*. Then, T is well-defined and continuous if and only if T* is well-defined and continuous. Consequently, from Proposition 3.2, it follows that T is positive (p, q)-summing, if and only if its adjoint T* E L(F*, X*) is strongly positive (q*,p*)-summing. Furthermore, ^|TK.q = ||^T* ||D + = ||^T || .

q*’p*

• ( 2) Let T E D+q(E,Y)• Then, by Proposition 3.2, the operator T : lq(E) —>l_p (Y) is continuous with |T||d+ = |T||. Using Theorem 1.3, and taking into account that the adjoint of the operator T : lq(E) —>l_p (Y) can be identified with the operator

T* : lp*,H(Y*) -^ lq* (E*) given by T*((у*Ы*) = (T*(у*)Ы*, we obtain T and T* are well-defined and continuous. Therefore, it follows from Proposition 2.1 that T is positive strongly (p, q)-summing if and only if its adjoint T* is positive (q*,p*)-summing, and ||T НD+q = |T ^Л^* = НГЦ. >

Corollary 3.4. Let 1 < q< p< ro.

• (1)    The operator T E L(E,Y) belongs to D+_q(E,Y) if and only if T** belongs to D+_q(E**,Y**). Furthermore,

Н^TНD^ = HT "HD+. •

• (2)    The operator T E L(X,F) belongs to Apq(X,F) if and only if T** belongs to A_p,_q(X**,F**). Furthermore,

НT Нл_p,_q = НT **K, •

We say that an operator T : E —> Y is called positive (p, q)-majorizing (see [10] for p = q) if there exists a constant C > 0, such that n                      q∗

IE l(T (z ),y')|^qJ  < C НШик,-

(3.5)

for all (zi)n=1 in Be and (y*)n=1 in (Y*)+. The space of all positive (p, q)-majorizing from E to Y is denoted by Yp,q(E, Y). This space becomes a Banach space with the norm || • I^p q given by the infimum of the constants C satisfying (3.5). In [10], the authors proved the duality relationships between positive p-summing operators and positive p-majorizing operators. It was known [3] that an operator T : X —>F is positive p-summing if and only if T* is positive strongly p*-summing. Similarly, an operator T : E —>Y is positive strongly p-summing if and only if T ^∗ is positive p^∗-summing. In the following, we directly prove that the concept of positive strongly p-summing and the concept of positive p-majorizing are equivalent.

Theorem 3.5. Let T : E —>Y be an operator. The following conditions are equivalent: (1) T is positive (p,q)-majorizing.

• (2)    T is positive strongly (p, q)-summing.

• <1 Suppose that T is positive (p, q)-majorizing, given any finite sequence (xi)n=1 in E and (y*)n₌₁in (Y*)⁺, we get

  D<^T(х,Ы)| = Ё Ix.||(^T (^) .У")

  
  

  n

  
  

  n

  
  

  i=1

  
  

  i=1

  
  

1                                       *\  1

C

(E lxI’) q (E IT (^) '»*)[p < IT!’>l(xi)-1Iql(y*)"=1 Ik

This implies that T is positive strongly (p, q)-summing and |T|d+ C IT|y_pq•

Conversely, assume that T is positive strongly (p, q)-summing. Let (zi )i=₁be a finite sequence in Be and (y*)^ in (Y*)⁺, we have

(ri

El (T (zi ),У* )|''

i=1

_1_ q∗

sup (AiJn^jeBiq

n

£ Ai(T(Zi),y** i=1

n

= sup     £(AiZi ),y*)< |T |d+     sup    I (Ai zi )i=1Iq I(y*)”=1Ip* ,_M

(Xi)n=_ieBlq i=1                          pq (Xi^EBlq

C IT Id+    sup   I (Ai )i=1|q |(y* )i=1lk_H= IT Id+ l(y*)i=1l_p^

q (XiY^eBlq                                    pq

This means that T is positive (p, q)-majorizing and ITIyp q CITI d+ • > p q

Corollary 3.6. T G L(E, Y) is positive p-majorizing if and only if T is positive strongly p-summing.

4. Positive Cohen (p, q)-Nuclear Operators

Cohen [6] introduced the concept of p-nuclear operators, which was extended to the Cohen (p, q)-nuclear operators by Apiola [14]. Let 1 < p,q C ж. An operator T G L (E, F) is Cohen (p, q)-nuclear if (T (x_n))neN* G l_p(F* whenever (xn)neN* G l_q,_u (E). We denote the space of Cohen (p, q)-nuclear operators by CN_p,_q(E, F). According to [6, 14], the following conditions are equivalent for a linear operator T G L (E, F):

T G CNp,q (E,F) . . T G L (lq,^ (E), Ip (F*),                  (4.1)

where T ((xn)neN*) = (T (xn))n_eN* for every (xn)neN* G lq,^ (E) •

In this section, we introduce the positive Cohen (p, q)-nuclear operators. For p = q, these operators are closely linked to positive strongly p-summing and positive p-summing operators, as stated in Kwapien’s Factorization Theorem (see [16, Proposition 2]). Here, we distinguish three cases.

Definition 4.1. Let 1 < q C p < oo and X, Y be Banach lattices, E and F be Banach spaces.

• (a)    An operator T from a Banach lattice X to a Banach space F is left positive Cohen (p, q)-nuclear if there exists a constant C > 0, such that for all (xi)П₌₁ C X+, we have (see [17])

  (4.2)

  
  

||(T(Xi))n₌₁||ip₍F) C Ch(Xi)n₌₁hq,_H.

• (b)    An operator T from a Banach space E to a Banach lattice Y is right positive Cohen (p, q)-nuclear if there exists a constant C > 0, such that for all (xi )n₌₁ C E, we have

  (4.3)

  
  

I(t(XiW=illin(y) C Cl(Xi)П=Л,_Ш

• (c)    An operator T from a Banach lattice X to a Banach lattice Y is positive Cohen (p, q)-nuclear if there exists a constant C > 0, such that for all (xi)n₌₁ C X, we have

l(T(Xi))n=1|ln(Y) C C||(Xi)n=1||q,H, see [10, Definition 3.1] for r = 1.

The class of all positive Cohen (p, q)-nuclear operators from X to Y (respectively X to F and E to Y) is denoted by CN +q(X, Y) (respectively CNPeft,+ (X,F) and CN rpigqht+(E,Y)).

We put |T|CN + = inf C.

The proof of the following results follows similar lines as in Proposition 3.2 and Proposition 3.22 in [17] and is omitted.

Proposition 4.2. Let 1 C q C p < o and X, Y be Banach lattices, E and F be Banach spaces.

• (1)    T G CN p^ft’⁺(X, F) if and only if T : l_q,|_ш|(X) —>ip{F) is a well-defined continuous linear operator [17, Proposition 3.22].

• (2)    T G CNPig^ht,+(E, Y) if and only if T : i_q,_u(E) —>Ip (Y) is a well-defined continuous linear operator.

• (3)    T G CN +q(X, Y) if and only if'T : i_q,m(X) —>Ip(Y) is a well-defined continuous linear operator.

A result by Apiola states that the adjoint of a Cohen (p, q)-nuclear linear operator is Cohen (q* ,p*)-nuclear linear operator. When p = q, this result appears in [6]. Utilizing Theorem 1.3, (1.1) and (1.2) and taking into account that the adjoint of the operators

T : iq^(X) -^ lP(F), T : iq,^(E) -^ ip(Y) and T : lq^(X) -^ ij(Y) can be identified with the operators

T* : ip*,. (F*) -^ in (X*), T* : iqM (Y*) -^    {E*) and T* : £p.M (Y*) -^ i^ (X*), defined as T*((xn)neN*) = (T*(xn))neN*, we extend this to positive Cohen (p, q)-nuclear operators.

Theorem 4.3. Let 1 C q C p < co and X, Y be Banach lattices, E and F be Banach spaces.

• (1)    The operator T belongs to CN ft^ (X,F) if and only if its adjoint T * belongs to CN ?^^⁺(F * , X *) • Furthermore,

||TH^ye^fS+ + — ||T* H^^ight,. .

CN p,q            CN _q∗,p∗

• (2)    The operator T belongs to CNPig^ht,+(E,Y) if and only if its adjoint T* CN^lfp+ (Y* ,E*). Furthermore,

  belongs

  
  

  to

  
  

  belongs

  
  

  to

  
  

^HT ^HCN;i^ght’+ ^{— H}T * ^HCN ^f+ •

• (3)    The operator T belongs to CN+_q(X, Y) if and only if its adjoint T* CN + _p* (Y*, X*) ■ Furthermore,

5. Tensor Characterizations

HTHCN + — HT HCN + ,• p,q                 q∗,p∗

Remark 4.4. In a recent paper [10, Definition 3.1], the authors introduced the concept of positive (p, q)-dominated, where 1/p +1/q — 1/r, defined between Banach lattices. Within this framework, both the Pietsch Domination Theorem and the Kwapien Factorization Theorem are established. This concept precisely aligns with the positive Cohen p-nuclear concept presented here when r — 1. Thus, by referring the reader to the papers [10, Theorem 3.3 and Theorem 3.7], we can also derive the well-known theorems, namely Pietsch’s Domination Theorem and Kwapien’s Factorization Theorem, for the other two concepts proposed here (for left and right positive Cohen p-nuclear). Notice that Kwapien’s Factorization Theorem ensures that positive Cohen p-nuclear are closely related to positive strongly p-summing and positive p-summing operators.

Now we are interested to characterize the aforementioned classes using abstract summability properties linked to the continuity of tensor product operators defined within vector-valued sequence spaces.

The Wittstock injective tensor product and Fremlin projective tensor product. For Banach lattices X and Y, let X 0 Y denote the algebraic tensor product of X and Y. For each u — ^П₌₁Xi 0 yi € X 0 Y, define T_u : X* ^ Y by T_u(x*) — £”=1 x*(xi,)yi for each x* G X*. The injective cone on X 0 Y is defined to be

n u — ^ Xi 0 yi G X 0 Y :

i=i

Tu(x*) G Y₊Vx* G X+

Wittstock [18, 19] introduced the positive injective tensor norm on X 0 Y as follows:

HuHi — inf {sup {HTv(x*)H : X* G Bx*} : v G Ci, u ± u G C^ •

.^^.

.^^.

Let X0iY denote the completion of X 0 Y with respect to ЦЛ^ Then X0iY with Ci as its positive cone is a Banach lattice (also see [20, Section 3.8 ]), called the Wittstock injective tensor product of X and Y. The projective cone on X 0 Y is defined to be

Cp — |^Xi 0 yi : Xi G X₊,yi G Y₊, n G n| •

Fremlin [21, 22] introduced the positive projective tensor norm on X 0 Y as follows:

{n                   n                           \ llullw = suP

^ ф(xi,Уi) : u = ^ Xi 0 yi G X 0 Y, ф G M>, i=1                    i=1                               J where M is the set of all positive bilinear functional ф on X x Y with ||ф|| C 1. Let X0fY denote the completion of X 0 Y with respect to || • || |n|. Then X0fY with Cp as its positive cone is a Banach lattice (also see [20, Sect. 3.8 ]), called the Fremlin projective tensor product of X and Y. Let p be real numbers, such that 1

(Pi) lu|^|(X) is isometrically lattice isomorphic to l_p0iX.

(P2) ip(X) is isometrically lattice isomorphic to l_p0fX.

----

,—-

p is X

Let i_p0_eE and i_p0_nE denote the Grothendieck injective and projective tensor product of with a Banach space E, respectively (see Ryan [23]). It is well known that the space lu^E) isometrically isomorphic to i_p0_eE whereas i_p(X) is isometrically isomorphic to i_p 0д_р (see [7, 12.9] and l_p(E) is isometrically isomorphic to i_p0_nE (see [6, Proposition 2.2.5

and Proposition 2.2.6], [24, Corrolary 3.9] and [25]). Given a linear operator T : X ^ Y, its associated tensor product operator 10 T : ip 0 X ^ ip 0 Y is defined by nn

£ ei 0 Xi := £fii 0 T(xi), i=1               i=1

and this map is clearly linear.

We apply now Theorem 2.2 and (Pi) to the class of positive (p, q)-summing operators to get new characterizations in terms of tensor product transformations.

Corollary 5.1. Let 1 < p < ro and T G L(X, F). The following conditions are equivalent:

• (1)    T is positive (p,q)-summing operator.

• (2)    The induced linear operator 10 T : i_q<0iX ^ i_p0_nF is continuous.

In this case \\T|лp,q = |I 0 T||.

According to (Pi) and Theorem 3.2, we obtain characterizations in terms of tensor product transformations for the class of positive strongly (p, q)-summing operators.

Corollary 5.2. Let 1 < p C ro and T G L(E, Y). The following properties are equivalent:

• (1)    T is positive strongly p-summing.

• (2)    The induced linear operator 10 T : i_p0^_pE ^ i_p0fY is continuous.

In this case ||T^D+q = |I 0 T||.

It is known from [6, Theorem 2.1.3] that T G L (E,F) is Cohen p-nuclear if and only if the mapping 10 T : £_p0_eE ^ i_p0_nF is continuous. Utilizing Proposition 4.2, (Pi) and (P2), we extend this result as follows.

Corollary 5.3. Let 1 C q C p < ro and X, Y be Banach lattices, E and F be Banach spaces.

• (ai ) T G L (X, F) is left positive Cohen (p, q)-nuclear if and only if the mapping 10 T : iq0iX ^ i_p0_nF is continuous.

• (a2 ) T G L (E, Y) is positive right Cohen (p, q)-nuclear if and only if the mapping 10 T : iq0_eE ^ i_p0fY is continuous.

• (аз ) = T G L (X, Y) is positive Cohen (p, q) -nuclear if and only if the mapping I 0 T : iq0iX ^ i_p0 fY is continuous.

Acknowledgments. We would like to thank the referees for their careful reading of the manuscript and for their valuable suggestions. Also, we acknowledge with thanks the support of the General Direction of Scientific Research and Technological Development (DGRSDT), Algeria.