VECTOR VALUED ORLICZ-LORENTZ SEQUENCE SPACES AND THEIR OPERATOR IDEALS

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S.A. MOHIUDDINE* AND KULDIP RAJ**

Abstract. In the present paper we shall introduce and study vector valued Orlicz- Lorentz sequence spaceslp,q,M,u,∆,A(X) on Banach spaceX with the help of an Musilak-Orlicz functionMand for different positive indicespandq. We also study their K¨othe and topological duals. Finally, we introduce the operator ideals with the help of the corresponding scalar sequence spaces and s-numbers.

1. Introduction and Preliminaries

Let X and Y be two sequence spaces and A = (ank) be an infinite matrix of real or complex numbers. Then we say that A defines a matrix mapping from X into Y if for every sequence x= (xk)^∞_k=0 ∈X, the sequence Ax={An(x)}^∞_n=0, the A-transform of x, is inY, where

(1.1) An(x) =

∞

X

k=0

ankxk (n∈N).

By (X, Y), we denote the class of all matricesAsuch thatA:X→Y. Thus,A∈(X, Y) if and only if the series on the right-hand side of (1.1) converges for eachn∈Nand every x∈X.

The matrix domainXA of an infinite matrixA in a sequence spaceX is defined by

(1.2) XA={x= (xk) :Ax∈X}.

The approach constructing a new sequence space by means of the matrix domain of a particular limitation method has recently been employed by several authors (see[20]).

The study of vector-valued sequence spaces (VVSS) was provoked by the work of A.

Grothendieck in [5]. Since then this theory has developed considerably in different direc- tions, (see [7], [14]) and references given therein.

An Orlicz function M : [0,∞) → [0,∞) is a continuous, nondecreasing and convex function such that M(0) = 0, M(x) > 0 forx > 0. Such function M always has the integral representation

M(x) = Z x

0

p(t)dt

where p(t), known as the kernel of M, is right continuous, non-decreasing function for t >0.It is clear that an Orlicz functionM is always increasing asM(x)→ ∞asx→ ∞.

Also tp(t)→ ∞ as t → ∞ and tp(t) = 0 for t = 0, [11]. However p(t)> 0 for t = 0 is equivalent to the fact that the Orlicz sequence spacelM is isomorphic tol1, [8]. Therefore, we presume here that the kernel p(t) has value 0 fort = 0 and obviously p(t)→ ∞ as t→ ∞.

1991Mathematics Subject Classification. Primary 46A45; 47B06; 47L20.

Key words and phrases. Lorentz sequence spaces;s-numbers of operators; Musielak-Orlicz function;

difference sequence spaces; operator ideals.

1

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For Orlicz functionM and kernel p,we defineq(s) = sup{t:p(t)≤s},s≥0. Clearly q possesses the same properties as p and the function N defined asN(x) =Rx

0 q(t)dt, is an Orlicz function. The functionsM andN are calledmutually complementary functions.

These functionsM andN satisfies Young’s inequality: xy≤M(x) +N(y), forxy≥0 and alsoM(αx)≤αM(x) for 0< α <1.

An Orlicz functionM is said to satisfy the ∆2-condition for smallxor at 0, if for each k >1, there existsRk>0 andxk>0 such that

M(kx)≤RkM(x) for allx∈(0, xk].

Suppose X andY be vector spaces over the same fieldKof real or complex numbers, generates a dual system hX, Yi with respect to the bilinear functional hx, yi. We shall denote the vector space of all sequences formed by the elements of X with respect to the operations of pointwise addition and scalar multiplication by Ω(X) and the space of all finitely non zero sequences from Ω(X) by φ(X). A vector-valued sequence space Λ(X) is a subspace of Ω(X) containing φ(X). The symbol δ_i^x exits for the sequence {0,0, ...,0, x,0,0, ...},wherexis placed at the ith coordinate. The notationx⁽ⁿ⁾denotes thenth section ofxgiven by{x1, x2, ..., xn,0,0, ...}.

A subset M of Λ(X) is said to be normal if for{xi} ∈ M and{αi} ∈K, with |αi| ≤ 1, i≥1, the sequence{αixi} ∈M. Thegeneralized K¨othe dual of Λ(X) is the space

Λ^×(Y) =n

y={yi} ∈Y :X

i≥1

|hxi, yii|<∞ for all{xi} ∈Λ(X)o .

The generalized K¨othe dual of Λ^×(Y) is denoted by Λ^××(X). The space Λ(X) is said to beperfect if Λ(X) = Λ^××(X).

A vector-valued sequence space Λ(X) equipped with a Hausdorff locally convex topology T is called

(i) a GK-space if the mapsPn,Λ(X): Λ(X)→ X, Pn,Λ(X)(x) = xn, for each n ≥1, are continuous;

(ii) aGAK-space if Λ(X) is a GK-space and for each{xi} ∈Λ(X), x⁽ⁿ⁾→xasn→ ∞, inT;

(iii) aGAD-spaceifx∈φ(X), for every x∈Λ(X) i.e. φ(X) = Λ(X).

Remark 1.1. Every perfect sequence space Λ(X) is normal [14].

Let us state here that if the dual system ishX, X^∗iwhereX is a Banach space andX^∗ is its topological dual, then we may interchangeably use the notationshx, fior f(x) for x∈X andf ∈X^∗ in the sequel.

We writewfor Ω(X),φforφ(X) andλfor Λ(X) if we takeX =K, the field of scalars,.

Ifen’s are thenth unit vectors inw, i.e. eⁿ={δnj}^∞_j=1, whereδnj is theKronecker delta, φis clearly the subspace ofwspanned byen’s,n≥1.

A sequence space λ is said to be symmetric if ασ = {ασ(i)} ∈ λ whenever α ∈ λ and σ∈Π, where Π is the collection of all permutations of N. The K¨othe dual λ^× of a symmetric sequence spaceλis symmetric [8].

Theδ-dual for scalar valued sequence spaceλis defined as λ^δ =n

α∈w:X

i≥1

|αiβρ(i)|<∞ for allβ∈λandρ∈Πo . λ^× coincides withλ^δ, ifλis symmetric.

We define

λ(X) =n

{xn}:xn ∈X, n≥1 and{kxnk} ∈λo

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for a scalar-valued sequence spaceλand a Banach spaceX. In case,λequipped with the normk.kλ, is a Banach space. Therefore,λ(X) is also a Banach space with respect to the norm

kxkλ(X)=k{kxnk}kλ (see [2], [7]).

As particular cases, we havel∞(X) forλ=l∞ andc0(X) corresponding toλ=c0. We define the set ˜lM(X) as

˜lM(X) ={x∈Ω(X) :X

i≥1

M(kxik)<∞}.

for a Banach spaceX corresponding to an Orlicz functionM. Thevector-valued Orlicz sequence space is defined as

lM(X) =n

x∈Ω(X) :X

i≥1

fi(xi) converges for all{fi} ∈˜lN(X^∗)o . for mutually complementary functionsM andN.

A corresponding way of defininglM(X) is lM(X) =n

x∈Ω(X) :X

i≥1

Mkxik ρ

<∞for someρ >0o . Two norms

kxk(M)= supn

X

i≥1

fi(xi) :X

i≥1

N(kfik)≤1o , and

kxkM = infn

ρ >0 :X

i≥1

Mkxik ρ

≤1o

; are equivalent onlM and hence we have

kxkM ≤ kxk_(M₎≤2kxkM, forx∈lM(X)

[21]. We shall writelM(X) aslM forX =K. If M satisfies ∆2-condition at 0 andM,N are mutually complementary Orlicz functions, then (lM)^×=lN [8].

AMusielak-Orlicz function M={Mn} is a sequence of Orlicz functions (see [4], [13]).

A Musielak-Orlicz function M is said to satisfy L1 condition if pn(x) ≥pn+1(x) for all x∈[0,∞) wherepn be the kernel ofMn for alln∈N. A convex modular%M onwfor a Musielak-Orlicz functionMis defined as

%M({αn}) = sup

σ∈Π

∞

X

n=1

Mn(ασ(n)).

Analogous to a convex modular%M, we define modular space as

λM={α={αn} ∈w:%M(βα)<∞, for someβ >0}.

This space becomes a normed space under the Luxemburg norm kαk= inf{β >0 :%M α

β ≤1}.

A modular sequence spaceλM is always a symmetric sequence space.

The decreasing rearrangement of the absolute values of a sequence α={αn} inl∞ is given by{tn(α)}, where

tn(α) = inf{ρ >0 : card{k:|αk|> ρ}< n}.

Here cardA denotes the cardinality of the setA. The sequence{tn(α)} satisfies the following properties [16]:

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(i)kαk∞=t1(α)≥t2(α)≥...≥0 forα∈l∞. (ii)tm+n−1(α+β)≤tm(α) +tm(β) forα, β∈l∞. (iii)tm+n−1(αβ)≤tm(α)tm(β) forα, β∈l∞. Hereαβ ={αnβn}.

Forx={xn} ∈l∞(X), we denote by

tn(x) =tn({xn}) =tn({kxnk}), n∈N. TheLorentz sequence space lp,q (0< p, q≤ ∞) is given by

lp,q={α={αn} ∈l∞:{n¹^p⁻¹^qtn(α)} ∈lq}.

Forα∈lp,q, let us consider the real-valued functionk.kp,q as follows

kαkp,q=









 n X

n≥1

(n¹^p⁻¹^qtn(α))^qo¹_q

for 0< p <∞, sup

n≥1

n¹^ptn(α) forq=∞.

For a convex modular%M defined on w, it has been proved in [4], that

(1.3) X

n≥1

Mn(tn(α)) =%M(α) forα∈wif and only ifMsatisfiesL1 condition.

(lp,q,k.kp,q) is Banach spaces for p ≥ q by equation (1.3). But for p < q, it is a quasi-Banach space. Further, they are symmetric sequence spaces [15].

We shall denote the Banach spaces over the complex fieldCbyX and Y and the class of all bounded linear maps fromX toY byL(X, Y) throughout the paper.

LetL be the class of all bounded linear operators between any pair of Banach spaces andw⁺be the class of sequences of non-negative real numbers. A mappings:L→w⁺ is called ans-number function if it satisfies the following conditions:

(i)kSk=s1(S)≥s2(S)≥...≥0, s(S) ={sn(S)}, S∈L, (ii)sn(S+T)≤sn(S) +kTkforS, T ∈L(X, Y) andn∈N,

(iii)sn(RST)≤ kRksn(S)kTk forT ∈L(X0, X), S∈L(X, Y), R∈L(Y, Y0) andn∈N, (iv) if rankS < n, thensn(S) = 0,

(v) if dimX ≥n, thensn(IX) = 1, where IX denotes the identity map ofX. If the condition (ii) is replaced by

(ii)⁰ sm+n−1(S+T)≤sm(S) +sn(T) forS, T ∈L(X, Y) andm, n= 1,2, ..., then thes-number function is calledadditive.

Ans-number function is calledmultiplicative if the condition (iii) is replaced by (iii)⁰ sm+n−1(RT)≤sm(R)sn(T) forR∈L(Y0, Y) andT∈L(X, Y0), m, n= 1,2, ...

We writeA(X, Y) =A∩L(X, Y) for a subsetAofL. Anoperator ideal is a collection ofAif it satisfies the following:

(i)Acontains all finite rank operators, (ii)T+S∈A(X, Y) forS, T ∈A(X, Y),

(iii) ifT ∈A(X, Y) andS ∈L(Y, Z), then ST ∈A(X, Z) and also if T ∈ L(X, Y) and S∈A(Y, Z), thenST ∈A(X, Z).

For the Banach spaces X andY the collectionA(X, Y) is called acomponent ofA.

A real valued functionf is said to be anideal quasi normiff defined on an operator ideal Aand satisfies the following properties:

(i) 0≤f(T)<∞, for eachT ∈Aandf(T) = 0 if and only ifT = 0,

(ii) there exists a constantσ≥1 such thatf(S+T)≤σ[f(S) +f(T)] forS, T ∈A(X, Y),

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whereA(X, Y) is any component ofA,

(iii) a)f(RS)≤ kRkf(S) forS∈A(X, Z), R∈L(Z, Y) and b)f(RS)≤ kSkf(R) forS∈L(X, Z), R∈A(Z, Y).

An operator ideal is said to be quasi-normed operator ideal if it is equipped with an ideal quasi-norm and a quasi-Banach operator ideal is a quasi-normed operator ideal of which each component is complete with respect to the ideal quasi-norm.

The notion of difference sequence spaces was introduced by Kızmaz [10] who studied the difference sequence spacesl∞(∆),c(∆) andc0(∆). The notion was further generalized by Et and C¸ olak [3] by introducing the spacesl∞(∆^m), c(∆^m) andc0(∆^m). Let mbe a non-negative integer, then forZ=c, c0andl∞, we have sequence spaces

Z(∆^m) ={x= (xk)∈w: (∆^mxk)∈Z},

where ∆^mx= (∆^mxk) = (∆^m−1xk−∆^m−1xk+1) and ∆⁰xk=xk for allk∈N, which is equivalent to the following binomial representation

∆^mxk=

m

X

v=0

(−1)^v m

v

xk+v.

Takingm= 1, we get the spaces studied by Et and C¸ olak [3]. For more details about this work one can refer to (see [1], [2], [9], [12], [16], [18], [19].)

2. The vector-valued sequence spaces lp,q,M,u,∆,A(X) andhp,q,M,u,∆,A(X) LetX be a Banach space. LetM= (Mk) be an Musielak-Orlicz function, that is,M is a sequence of Orlicz functions,u= (uk) be a sequence of strictly positive real numbers andA = (ank) be a nonnegative two-dimensional bounded-regular matrix. In this paper we define the following classes of sequences:

lp,q,M,u,∆,A(X) =

x={xk} ∈l∞(X) :X

k≥1

uk

hMk

kA k^p¹⁻¹^qtk(∆^mxk)k ρ

i<∞,

for someρ >0

.

hp,q,M,u,∆,A(X) =

x={xk} ∈l∞(X) :X

k≥1

uk

hMk

kA k^p¹⁻¹^qtk(∆^mxk)k δ

i<∞, for allδ >0

.

Forx∈lp,q,M,u,∆,A(X), we define kxkp,q,M,u,∆,A(X) = inf

ρ >0 :X

k≥1

uk

hMk

i≤1

. If we takeM(x) =xinlp,q,M,u,∆,A(X) andhp,q,M,u,∆,A(X), then we have the following spaces:

lp,q,u,∆,A(X) =

x={xk} ∈l∞(X) :X

k≥1

uk

hkA k^p¹⁻¹^qtk(∆^mxk)k ρ

i<∞, for someρ >0

,

hp,q,u,∆,A(X) =

x={xk} ∈l∞(X) :X

k≥1

uk

hkA k¹^p⁻¹^qtk(∆^mxk)k δ

i<∞, for allδ >0

.

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Letu= (uk) = 1, for allk∈N. Then the spaces lp,q,M,u,∆,A(X) andhp,q,M,u,∆,A(X) are reduced tolp,q,M,∆,A(X) andhp,q,M,∆,A(X), respectively, as follow:

lp,q,M,∆,A(X) =

x={xk} ∈l∞(X) :X

k≥1

hMk

kA k¹^p⁻¹^qtk(∆^mxk)k ρ

,

hp,q,M,∆,A(X) =

x={xk} ∈l∞(X) :X

k≥1

hMk

kA k¹^p⁻¹^qtk(∆^mxk)k δ

i<∞, for allδ >0

. If we takeA= (C,1) inlp,q,M,u,∆,A(X) andhp,q,M,u,∆,A(X), then we have the following spaces:

lp,q,M,u,∆(X) =

x={xk} ∈l∞(X) :X

k≥1

uk

hMk

kk¹^p⁻¹^qtk(∆^mxk)k ρ

,

hp,q,M,u,∆(X) =

x={xk} ∈l∞(X) :X

k≥1

uk

hMk

kk¹^p⁻¹^qtk(∆^mxk)k δ

i<∞, for allδ >0

. If we takeA= (C,1) andM(x) =xinlp,q,M,u,∆,A(X) andhp,q,M,u,∆,A(X), then we have the following spaces:

lp,q,u,∆(X) =

x={xk} ∈l∞(X) :X

k≥1

uk

hkk¹^p⁻¹^qtk(∆^mxk)k ρ

,

hp,q,u,∆,(X) =

x={xk} ∈l∞(X) :X

k≥1

uk

hkk¹^p⁻¹^qtk(∆^mxk)k δ

i<∞, for allδ >0

. If we take (Mk) = M, A = I, (uk) = 1 for all k ∈ Nand m = 0, then we get the analogous of the spaces defined by Gupta and Bhar [6]. The main aim of this paper is to study the vector-valued Orlicz-Lorentz sequence spaces. We study their structural properties and investigate cross and topological duals of these spaces. Finally we prove that the operator ideals defined with the help of scalar valued sequence spaceslp,q,M,u,∆,Aand additives-numbers are quasi-Banach operator ideals forp < qand Banach operator ideals forp≥q.

Main Results

Theorem 2.1. Let M= (Mk)be a Musielak-Orlicz function ,u= (uk)be a sequence of strictly positive real numbers and A = (ank) be a nonnegative two-dimensional bounded- regular matrix. Then the space lp,q,M,u,∆,A(X) equipped with k.kp,q,M,u,∆,A is a quasi- Banach space forp < q and Banach space forp≥q. Further for x∈lp,q,M,u,∆,A(X), we have

(2.1) X

k≥1

uk

hMk

kA k^p¹⁻¹^qtk(∆^mxk)k kxkp,q,M,u,∆,A

i≤1.

Proof. We can easily show that lp,q,M,u,∆,A(X) is a vector space with usual coordinate wise addition and scalar multiplication. To show that k.kp,q,M,u,∆,A is a quasi-norm, let kxkp,q,M,u,∆,A ≥ 0 for each x ∈ lp,q,M,u,∆,A(X) and kxkp,q,M,u,∆,A = 0 for x = 0.

Suppose thatkxkp,q,M,u,∆,A= 0 for somex={xk} ∈lp,q,M,u,∆,A(X) and for givenε >0, we can findρ >0 such thatρ < εand

X

k≥1

uk

hMk

i≤1.

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Whenx6= 0, we get kxk0k 6= 0 for somek0∈Nand so tk1(∆^mxk) =k∆^mxk0k,for some k1∈Nimplies

uk

hMk

kA k

1 p−¹_q

1 tk1(∆^mxk)k ε

i≤uk

hMk

kA k

1 p−¹_q

1 tk1(∆^mxk)k ρ

i≤1 for anyε >0. We get a contradiction to the fact sox= 0.

Now to prove triangular-type inequality, considerx={xk}andy={yk} ∈lp,q,M,u,∆,A(X), Thus for anyε >0, there existρ1,ρ2>0 such that

ρ1<kxkp,q,M,u,∆,A+ε

2 with X

k≥1

uk

hMk

kA k¹^p⁻¹^qtk(∆^mxk)k ρ1

i≤1, and

ρ2<kykp,q,M,u,∆,A+ε

2 with X

k≥1

uk

hMk

kA k^p¹⁻¹^qtk(∆^myk)k ρ2

i≤1.

If ¹_p−¹_q >0, then via properties (i) and (ii) of{tk(∆^mxk)}, we get X

k≥1

uk

hMk

kA k¹^p⁻¹^qtk(∆^m(xk+yk))k 2^p¹⁻¹^q⁺¹(ρ1+ρ2)

i

= X

k≥1

uk

hMk

kA(2k)^p¹⁻¹^qt2k(∆^m(xk+yk))k 2^p¹⁻¹^q⁺¹(ρ1+ρ2)

i

+ X

k≥1

uk

hMk

kA(2k−1)^p¹⁻¹^qt2k−1(∆^m(xk+yk))k 2¹^p⁻¹^q⁺¹(ρ1+ρ2)

i

≤ 2X

k≥1

uk

hMk

kA k^p¹⁻¹^q(tk(∆^mxk) +tk(∆^myk))k 2(ρ1+ρ2)

i

≤ X

k≥1

uk

hMk

ρ1

ρ1+ρ2

+ ρ2

ρ1+ρ2

kA k¹^p⁻¹^qtk(∆^myk)k ρ2

i≤1 Hence,

kx+ykp,q,M,u,∆,A ≤ 2¹^p⁻¹^q⁺¹(ρ1+ρ2)

≤ 2¹^p⁻¹^q⁺¹(kxkp,q,M,u,∆,A+kykp,q,M,u,∆,A+ε).

Now we prove the completeness of the space (lp,q,M,u,∆,A(X),k.kp,q,M,u,∆,A),let{xk}be a Cauchy sequence inlp,q,M,u,∆,A(X) as xk ={xⁿ_k}n≥1, k ∈ N. Hence for ε > 0, there existsk0∈Nsuch that

kxk+j−xkkp,q,M,u,∆,A= infn

ρ >0 :X

n≥1

un

hMn

kA n¹^p⁻¹^qtn(∆^m(xk+j−xk))k ρ

i≤1o

< ε

for eachk≥k0and each j∈N. Thus X

n≥1

hkA n¹^p⁻^q¹tn(∆^m(xk+j−xk))k ε

i≤1 for allk≥k0, j∈N.

=⇒ n

{A n¹^p⁻¹^qtn(∆^m(xk+j−xk))/ε} : n ∈ No

is a bounded set forj ∈ Nand for all k≥k0.Therefore {xⁿ_k}is a Cauchy sequence inX,for eachn∈Nand so converges tozn.

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Letz={zn}Then tn(∆^m(xk+j−xk))→tn(z−xk) asj→ ∞ and hence by continuity ofM,

X

n≥1

un

hMn

kA n¹^p⁻¹^qtn(∆^m(z−xk))k ε

i≤1 for allk≥k0.

This implies thatz∈lp,q,M,u,∆,A(X) andkz−xkkp,q,M,u,∆,A→0 ask→ ∞.

Equation (2.1) is directly from the definition of the quasi-normk.kp,q,M,u,∆,A.This com-

pletes the proof.

Theorem 2.2. Let M= (Mk) be a Musielak-Orlicz function, u= (uk) be a sequence of strictly positive real numbers and A = (ank) be a nonnegative two-dimensional bounded- regular matrix. Thenhp,q,M,u,∆,A(X) is a closed subspace oflp,q,M,u,∆,A(X). Moreover, if M= (Mk)satisfies∆2-condition at0 thenlp,q,M,u,∆,A(X) =hp,q,M,u,∆,A(X).

Proof. First of all it is without a doubt thathp,q,M,u,∆,A(X) is a subspace oflp,q,M,u,∆,A(X).

Now we prove that hp,q,M,u,∆,A(X) is closed in lp,q,M,u,∆,A(X). Suppose x = {xk} ∈ hp,q,M,u,∆,A(X), the closure of hp,q,M,u,∆,A(X) inlp,q,M,u,∆,A(X). So there exists a sequence {yk}={{y_kⁿ}} ∈hp,q,M,u,∆,A(X),k≥1 and we havekyk−xkp,q,M,u,∆,A→0 as k→ ∞.

Take anyδ >0. Thus forδ1= min{2^p¹⁻¹^qδ, δ}, we getk0∈Nsuch that (2.2) kyk−xkp,q,M,u,∆,A< δ1

2 for allk≥k0. When ¹_p−¹_q ≥0,we have

X

n≥1

un

hMn

kA n¹^p⁻¹^qtn(∆^mx)k δ

i

≤ 2X

n≥1

un

hMn

kA n¹^p⁻¹^q tn(∆^m(x−yk0)) +tn(∆^myk0) k δ1

i

≤ X

n≥1

un

hMn

kA n¹^p⁻¹^q tn(∆^m(x−yk0)) k δ1/2

i

+ X

n≥1

un

hMn

kA n¹^p⁻¹^qtn(∆^myk0)k δ1/2

i

≤ X

n≥1

un

hMn

kA n¹^p⁻¹^q tn(∆^m(x−yk0)) k kx−yk0kp,q,M,u,∆,A

i

+ X

n≥1

un

hMn

i

< ∞.

In the case when ¹_p−¹_q <0,we get

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X

n≥1

un

hMn

kA n¹^p⁻¹^qtn(∆^mx)k δ

i

≤ X

n≥1

un

hMn

kA n¹^p⁻¹^q tn(∆^m(x−yk0)) k δ1/2

i

+ X

n≥1

un

hMn

i<∞.

by equation (2.2). Clearly x∈ hp,q,M,u,∆,A(X) and so the subspace hp,q,M,u,∆,A(X) is closed. Now we suppose thatMsatisfies ∆2-condition at 0. Letx∈lp,q,M,u,∆,A(X), we have

X

k≥1

uk

hMk

i<∞ for someρ0>0.

To show thatx∈hp,q,M,u,∆,A(X), choose anyη >0. Ifη≥ρ0, then X

k≥1

uk

h Mk

kA k¹^p⁻¹^qtk(∆^mxk)k η

i

<∞.

Now presumeη < ρ0 and supposeK= ^ρ_η⁰. SinceMsatisfies the ∆2-condition, so we can findRK>0 andxK>0 such thatM(Kx)≤RKM(x) for allx∈(0, xK]

=⇒ X

k≥k0

uk

hMk

kKA k¹^p⁻¹^qtk(∆^mxk)k ρ0

i

≤RK

X

k≥k0

uk

hMk

kA k¹^p⁻^q¹tk(∆^mxk)k ρ0

i<∞

for somek0∈N. Hence X

k≥1

uk

hMk

kA k^p¹⁻¹^qtk(∆^mxk)k η

i<∞, for anyη >0

and sox∈hp,q,M,u,∆,A(X), we havehp,q,M,u,∆,A(X) =lp,q,M,u,∆,A(X).

Proposition 2.3. Let M= (Mk) be a Musielak-Orlicz function,u= (uk) be a sequence of strictly positive real numbers andA= (ank)be a nonnegative two-dimensional bounded- regular matrix. IfY =hp,q,M,u,∆,A(X)∩c0(X),0< p, q≤ ∞. ThenY equipped with the subspace topology ofhp,q,M,u,∆,A(X)is a GAD-space.

Proof. Obviouslyφ(X)⊂Y. Supposex∈Y.Now for anyε >0, we can findk0∈N, such that

X

k≥k0

uk

hMk

kA k^1/p−1/qtk(∆^mxk)k ε

i≤1.

LetIk ={i∈N:kxik> ¹_k}, k∈Nandvk= X

i∈Ik

δ_i^xⁱ. Sincex∈c0(X),Ik is finite and so vk ∈φ(X).Setnk= cardIk. Then takem0∈Nsuch that

X

k≥m0

uk

hMk

kA k^1/p−1/qtk(∆^mxk)k ε

i≤ 1 2.

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Takek so large that

1 k

m0

X

i=1

uk

hMk

kA i^1/p−1/qk ε

i≤1 2. Thus,

X

i≥1

uk

hMk

kA i^1/p−1/qtn_k+i(∆^mxk)k ε

i

≤ 1 k

m0

X

i=1

uk

hMk

kA i^1/p−1/qk ε

i+ X

i≥m0+1

uk

hMk

kA k^1/p−1/qti(∆^mxk)k ε

i

≤ 1 2 +1

2 = 1.

This implies that kx−vkkp,q,M,u,∆,A ≤ ε for sufficiently large k. Hence Y is a GAD-

space.

Proposition 2.4. LetM= (Mk)be a Musielak-Orlicz function satisfying∆2-condition at 0,u= (uk)be a sequence of strictly positive real numbers andA= (ank)be a nonnegative two-dimensional bounded-regular matrix. If lp,q,M,u,∆,A(X) ⊂ c0(X), 0 < p ≤ q ≤ ∞.

Thenlp,q,M,u,∆,Ais a GAD-space.

Remark 2.5. It is very motivating to know whether the spacehp,q,M,u,∆,A(X) is a GAK- space, this means that the k^th section x^(k) = {x1, x2, ..., xk,0,0,0,0, ...} of an element x = {xi} of hp,q,M,u,∆,A converges to x with respect to its quasi-norm. Whenever, if p, q >0 with ¹_p− ¹_q ≥ 0 and x∈ hp,q,M,u,∆,A(X) such that kx1k > kx2k > kx3k> ..., thentk(x) =kxkkand in this case, one can easily show thatkx−x^(k)kp,q,M,u,∆,A→0 as k→ ∞.

3. Duals of the space lp,q,M,u,∆,A(X), 1≤p≤q≤ ∞

Suppose that the spaceslp,q,M,u,∆,A(X) are symmetric sequence spaces, since the decreasing rearrangement ofxwould be the same as that ofxπ for any permutationπofN and M= (Mk) is an increasing function. Thus theδ-dual of the scalar valued sequence spacelp,q,M,u,∆,A would coincide with its cross-dual.

Theorem 3.1. Let M= (Mk)and N = (Nk)be two mutually complementary Musielak- Orlicz functions such thatMsatisfies∆2-condition at 0,u= (uk)be a sequence of strictly positive real numbers and A = (ank) be a nonnegative two-dimensional bounded-regular matrix. Then(lp1,q1,M,u,∆,A)^×⊇lp2,q2,N,u,∆,A, where1/p1+1/p2= 1and1/q1+1/q2= 1.

Moreover,(lp1,q1,M,u,∆,A)^×=lp2,q2,N,u,∆,A when1/p1−1/q1≥0.

Proof. To show that lp2,q2,N,u,∆,A⊂(lp1,q1,M,u,∆,A)^×,suppose β ∈lp2,q2,N,u,∆,A. Then, we have

X

k≥1

uk

hNk

kA k^p¹²⁻^q¹²tk(∆^mβ)k δ0

i<∞, for someδ0>0.

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Let α ∈ lp1,q1,M,u,∆,A. Then X

k≥1

uk

hMk

kA k^p¹¹⁻^q¹¹tk(∆^mα)k ρ

i < ∞ for all ρ > 0.

Thus,

X

k≥1

|αkβk| ≤ X

k≥1

tk(∆^mα)tk(∆^mβ)

≤ X

k≥1

uk

hMk

kA k^p1¹⁻^q1¹tk(∆^mα)k 1/δ0

i

+ X

k≥1

uk

hNk

kA k^p¹²⁻^q¹²tk(∆^mβ)k δ0

i<∞.

Hence β ∈ (lp1,q1,M,u,∆,A)^×. Now to prove (lp1,q1,M,u,∆,A)^× = lp2,q2,N,u,∆,A. Suppose β ∈ (lp1,q1,M,u,∆,A)^×, then X

i≥1

|αiβi| < ∞, ∀{αi} ∈ lp1,q1,M,u,∆,A. Since lp1,q1,M,u,∆,A

and (lp1,q1,M,u,∆,A)^× both are symmetric sequence spaces,{tk(∆^mα)} ∈lp1,q1,M,u,∆,Afor α ∈ lp1,q1,M,u,∆,A and {tk(∆^mβ)} ∈ (lp1,q1,M,u,∆,A)^× for β ∈ (lp1,q1,M,u,∆,A)^×. Hence X

k≥1

tk(∆^mα)tk(∆^mβ)<∞for allα∈lp1,q1,M,u,∆,A.

Again ifγ∈lM, then{tk(∆^mγ)} ∈lMaslMis symmetric and normal and so{Ak^p¹²⁻^q¹²tk(∆^mγ)}

∈lp1,q1,M,u,∆,A.Hence X

k≥1

hkA k^p¹²⁻^q¹²tk(∆^mγ)tk(∆^mβ)ki

<∞, for allγ∈lM

=⇒ {Ak^p¹²⁻^q¹²tk(∆^mβ)} ∈l^×_M=lN =⇒β∈lp2,q2,N,u,∆,A.

Therefore, (lp1,q1,M,u,∆,A)^×=lp2,q2,N,u,∆,A. Proposition 3.2. Let M= (Mk) be a Musielak-Orlicz function,u= (uk) be a sequence of strictly positive real numbers andA= (ank)be a nonnegative two-dimensional bounded- regular matrix. For positive realsp1, p2, q1, q2with 1/p1+ 1/p2= 1,1/q1+ 1/q2= 1 such thatq1< p1, the spaceslp1,q1,M,u,∆,Aare perfect sequences spaces.

Proof. Infact in this case (lp2,q2,N,u,∆,A)^×=lp1,q1,M,u,∆,Aandlp2,q2,N,u,∆,A⊆(lp1,q1,M,u,∆,A)^×. Solp1,q1,M,u,∆,A⊂(lp1,q1,M,u,∆,A)^××⊂(lp2,q2,N,u,∆,A)^× =lp1,q1,M,u,∆,A. Proposition 3.3. Let M= (Mk)be an Musielak-Orlicz function satisfying∆2-condition at 0,u= (uk)be a sequence of strictly positive real numbers andA= (ank)be a nonnegative two-dimensional bounded-regular matrix. LetX be a Banach space andp1, p2, q1, q2are such that1/p1+1/p2= 1,1/q1+1/q2= 1and1/p1−1/q1>0. Then(lp1,q1,M,u,∆,A(X))^× = lp2,q2,N,u,∆,A(X^∗).

Proof. One can easily prove it by using Theorem 3.1, so we omit the proof.

Theorem 3.4. Let M= (Mk) be a Musielak-Orlicz function, u= (uk) be a sequence of strictly positive real numbers and A = (ank) be a nonnegative two-dimensional bounded- regular matrix. Suppose p1, p2, q1, q2 are real numbers with 1 < p1, q1, p2, q2 < ∞ and 1/p1+ 1/p2 = 1, 1/q1+ 1/q2 = 1. Then the dual of lp1,q1,M,u,∆,A(X) is topologically isomorphic to lp2,q2,N,u,∆,A(X^∗) if and only if the sequence {fi} ∈ lp2,q2,N,u,∆,A(X^∗) is identified with the linear functionalF given by

(3.1) F({xi}) =X

i≥1

hxi, fii, for each {xi} ∈lp1,q1,M,u,∆,A(X).

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Proof. Subsequently for{fi} ∈lp2,q2,N,u,∆,A(X^∗),we define a linear functionalF on lp1,q1,M,u,∆,A(X) as in (+) where convergence of the series is guaranted by proposition 3.3. Fork∈N, let

Fk({xi}) =

k

X

i=1

hxi, fii, {xi} ∈lp1,q1,M,u,∆,A(X).

Obviously,{Fk} is a sequence of continuous linear functionals on lp1,q1,M,u,∆,A(X) con- verging pointwise toF. Thus,F is continuous by Banach-Steinhaus Theorem, [17]. Hence, F ∈(lp1,q1,M,u,∆,A(X))^∗. Nextly, forx∈lp1,q1,M,u,∆,A(X), we get

|f(x)| ≤ X

i≥1

|hxi, fii|

≤ X

i≥1

ti(x)ti(f)

≤ kfkp2,q2,N,u,∆,A

X

i≥1

ui

hMi

|A(i^1/p¹^−1/q¹ti(∆^mx))kk(i^1/p²^−1/q²ti(∆^mf))k kfkp2,q2,N,u,∆,A

i

≤ kfkp2,q2,N,u,∆,AkuiA{i^1/p¹^−1/q¹ti(∆^mx)}k(M), since X

i≥1

ui

hNi

kA i^1/p²^−1/q²ti(∆^mf)k kfkp2,q2,N,u,∆,A

i≤1. Therefore,

|F(x)| ≤2kfkp2,q2,N,u,∆,Akxkp1,q1,M,u,∆,A. for anyx∈lp1,q1,M,u,∆,A(X). Thus,

(3.2) kfk ≤2k{fi}kp2,q2,N,u,∆,A.

Conversely, suppose F ∈(lp1,q1,M,u,∆,A(X))^∗. Define fi ∈ X^∗, i ∈ Nas fi(x) = F(δ_i^x).

Now to prove {fi} ∈lp2,q2,N,u,∆,A(X^∗) we choose {αi} ∈lp1,q1,M,u,∆,A. Take {xi} ⊆ X withkxik= 1 andkfik< fi(xi)+1/2ⁱ, for alli∈N. Let{βi} ⊂Cbe such that|fi(αixi)|= fi(αiβixi), ∀i ∈ N. Obviously, |βi| = 1, ∀i ∈ N, and so {αiβixi} ∈ lp1,q1,M,u,∆,A(X).

Suppose

X

i≥1

|αi|kfik < X

i≥1

fi(αiβixi) +X

i≥1

αi

2ⁱ

= X

i≥1

F(δ^αⁱ^βⁱ^xⁱ) +K

= lim

k→∞

k

X

i=1

F(δ^αⁱ^βⁱ^xⁱ) +K

= F({δ^αⁱ^βⁱ^xⁱ}) +K where K = X

i≥1

αi

2ⁱ. Therefore, X

i≥1

|αi|kfik < ∞, ∀{αi} ∈ lp1,q1,M,u,∆,A and hence {fi} ∈lp2,q2,N,u,∆,A(X^∗) by Theorem 3.1.

To prove thatFhas the form as given in equation (3.1), suppose for{xi} ∈lp1,q1,M,u,∆,A(X) X

i≥1

|hxi, fii| = X

i≥1

|F(δ^xⁱ)|

= lim

k→∞

k

X

i=1

F(δ^βⁱ^xⁱ) =F({βixi})

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whereβi are taken as above. ThusX

i≥1

|hxi, fii|<∞. Hence,

(3.3) X

i≥1

|hxi, fii| is unconditionally convergent.

Now if 0< p1< q1≤ ∞, lp1,q1,M,u,∆,A(X) is a GAD-space. We writeti(x) =kxφ(i)k for someφ∈πanduk =

k

X

i=1

δ_φ^x_(i)^φ⁽ⁱ⁾, fork∈N.

Therefore uk∈φ(X) andkx−ukkp1,q1,M,u,∆,A→0 ask→ ∞by proposition 3.3. Then, F({xi}) =F( lim

k→∞uk) =X

i≥1

hxi, fii by equation (3.3).

Hence the mappingR : lp2,q2,N,u,∆,A(X^∗)→(lp1,q1,M,u,∆,A(X))^∗ defined by R(f) = F, with f ={fi}, fi(x) = F(δ_i^x), i∈ N is a topological isomorphism from equations (3.2), (3.1) and the open mapping theorem, [17]. This completes the proof.

4. The Operator Ideals L^(s)p,q,M,u,∆,A,0< p, q≤ ∞ LetM= (Mk) be a Musielak-Orlicz function and X, Y are Banach spaces.

Definition 4.1. Let T : X →Y be a bounded linear operator. Then T is said to be of typelp,q,M,u,∆,A if{sk(T)} ∈lp,q,M,u,∆,A. We shall denote the set of all above mappings byL^(s)p,q,M,u,∆,Awhere,

L^(s)p,q,M,u,∆,A={T ∈L:{sk(T)} ∈lp,q,M,u,∆,A}.

We define the norm for anyT ∈L^(s)p,q,M,u,∆,Aas kTkp,q,M,u,∆,A= infn

ρ >0 :X

k≥1

uk

hMk

kA k¹^p⁻¹^q∆^msk(T)k ρ

i≤1o .

Theorem 4.2. Let M= (Mk) be a Musielak-Orlicz function, u= (uk) be a sequence of strictly positive real numbers and A = (ank) be a nonnegative two-dimensional bounded- regular matrix. Then for p < q, L^(s)p,q,M,u,∆,A equipped with k.kp,q,M,u,∆,A is a quasi- Banach operator ideal and forp≥q it is a Banach ideal .

Proof. To show that L^(s)p,q,M,u,∆,A is an operator ideal, firstly note that all finite rank operators are contained inL^(s)p,q,M,u,∆,A, since sk(T) = 0 fork≥k0if rank T < k0. ForT1, T2∈L^(s)p,q,M,u,∆,A(X, Y), we have

X

k≥1

uk

hMk

kA k¹^p⁻¹^q∆^msk(T1)k ρ1

i<∞, for someρ1>0.

X

k≥1

uk

hMk

i<∞, for someρ2>0.

Firstly we consider the condition when ¹_p−¹_q ≥0

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X

k≥1

uk

hMk

kA k¹^p⁻¹^q∆^msk(T1+T2)k 2^p¹⁻¹^q⁺¹(ρ1+ρ2)

i

≤ X

k≥1

ρ1

ρ1+ρ2

uk

hMk

i

+ X

k≥1

ρ2

ρ1+ρ2

uk

hMk

i<∞.

Again if ¹_p−¹_q <0, then X

k≥1

uk

hMk

kA k¹^p⁻¹^q∆^msk(T1+T2)k (ρ1+ρ2)

i

≤ X

k≥1

ρ1

ρ1+ρ2uk

hMk

i

+ X

k≥1

ρ2

ρ1+ρ2

uk

hMk

i<∞.

This implies thatT1+T2∈L^(s)p,q,M,u,∆,A(X, Y).

Now, we want to show that for T ∈L^(s)p,q,M,u,∆,A(E, F), R∈ L(F, Y) andS ∈L(X, E), RT S ∈L^(s)p,q,M,u,∆,A(X, Y).Thus forT ∈L^(s)p,q,M,u,∆,A(E, F), we have

X

k≥1

uk

hMk

kA k^p¹⁻¹^q∆^msk(T)k ρ0

i<∞,

for someρ0>0 and hence X

k≥1

uk

hMk

kA k^p¹⁻¹^q∆^msk(RT S)k kRkkSkρ0

i<∞

by the property (iii) ofs-number function. ThusRT S ∈L^(s)p,q,M,u,∆,A(X, Y). Therefore, L^(s)p,q,M,u,∆,A is an operator ideal.

To prove the functionk.kp,q,M,u,∆,Ais a quasi-norm/ norm nature defined onL^(s)p,q,M,u,∆,A

is similar to the one defined onLp,q,M,u,∆,A(X) and so excluded.

To prove the completeness, suppose{Tk}is a Cauchy sequence in component ofL^(s)p,q,M,u,∆,A(X, Y) ofL^(s)p,q,M,u,∆,A. Thus forε >0, there existsk0∈Nsuch that

kTk+j−Tkkp,q,M,u,∆,A< ε for allk≥k0andj∈N This implies that there existsρ >0 such thatρ < εand

(4.1) X

n≥1

un

hMn

kA n¹^p⁻¹^q∆^msn(Tk+j−Tk)k ε

i≤1, for allk≥k0, j∈N.

Thus,{^{A n}

1 p−1

q∆^msn(Tk+j−Tk)

ε ;n≥1}is a bounded sequence for eachk≥k0 andj∈N. Therefore, for some constantK >0,we get

kTk+j−Tkk< εK for allk≥k0, j∈N.