Matrix Operator of Musielak-ϕ Function Sequence Space

M. Oﬁe

and E. Herawati

1∗

Department of Mathematics, Universitas Sumatera Utara, Medan, Indonesia

Keywords:

Matrix Operator, Musielak-ϕ Function, Banach Space.

Abstract:

Let X be a Banach space and Φ = {ϕ

} be a sequence of ϕ-function called Musielak-ϕ-function. In this

work, we introduce a vector valued sequence space generated by Musielak-ϕ function, `

∃

(X,Φ), and study

the matrix operator from the space `

(X) into the space `

∃

(X,Φ).

1 INTRODUCTION

Let X = (X,k · k

) be a Banach space. We de-

note space Ω(X) as a colection of all X-valued se-

quences. For every natural numbers i, a sequence

x = (x(i)) ∈ Ω(X) means x(i) in Banach space X.

Any linear subspace E ⊂ Ω(X) is called X-valued se-

quence space.

A function ϕ that deﬁned from R into R

∪ {0} is

called a ϕ-function if ϕ is even, continuous, vanishing

at zero, and increasing (Rao and Ren, 2002). For any

ϕ-function and for every real number x, if there exists

a constant K > 0 such that ϕ(2x) ≤ Kϕ(x), then ϕ

is called satisfy ∆

-condition. For a ϕ-function, ϕ,

satisﬁed ∆

-condition, the following space, denoted

by `

∃

(ϕ) is a generalization of Orlicz sequence space

(Kolk, 2015) i. e.

∃



x = (x

) : x

∈ R and ϕ





∈ `

for some ρ > 0





x = (x

) : x

∈ R and

∞

∑

i=1





< ∞

for some ρ > 0



respected to the following norm

||x||

= inf



ρ > 0 : ϕ





≤ 1



The space `

∃

(ϕ) is a Banach space.

An X-valued sequence space deﬁned by arith-

metic mean of ϕ-function has been introduced (Gul-

tom and Herawati, 2018). They studied some topo-

logical properties using paranorm and inclusion rela-

tions of this space. For E be a Riesz space (Herawati

et al, 2016) introduced E-valued sequence space de-

ﬁned by an order ϕ-function and proved that spaces

are ideal Banach lattices using lattice norm. In (Mur-

saleen, 2013) introduced some sequence spaces de-

ﬁned by a Musielak-Orlicz function and studied some

topological properties respected to n-norm and proved

some inclusion relations between these spaces. Us-

ing Musielak-Orlicz function to generated sequence

spaces equipped with the Luxemburg norm, pack-

ing constant of these space has been studied (Hudzik

et al, 1994). (Suantai, 2003) considered the char-

acterization problem of inﬁnite real matrices opera-

tors `(X, p) for p = (p

) is a bounded sequence with

0 ≤ p

≤ 1 for all natural number i and two others se-

quence space into the Orlicz sequence space, `

, for

M is an Orlicz function.

A matrix operator is an operator from any se-

quence space X to another sequence space Y by us-

ing inﬁnite real matrix A = (a

). i.e. an operator

A : X → Y with

Ax = (A

(x)) ∈ Y, for any x ∈ X

for every x ∈ X and for every natural number n

(x) =

∑

k≥1

< ∞.

Collection of matrix operator A : X → Y denoted by

(X,Y ). The sequence e

(k)

is a sequence which only

non zero-term is 1 in k

entry for every natural num-

ber k.

Let `

∃

(X,Φ) for X and Φ = {ϕ

} be a Banach

space and a Musielak-ϕ function, respectively, stud-

ied with luxemburg norm (Oﬁe and Herawati, 2018),

Oﬁe, M. and Herawati, E.

Matrix Operator of Musielak-j Function Sequence Space.

DOI: 10.5220/0010080309790981

In Proceedings of the International Conference of Science, Technology, Engineering, Environmental and Ramiﬁcation Researches (ICOSTEERR 2018) - Research in Industry 4.0, pages

979-981

ISBN: 978-989-758-449-7

979

i.e.

∃

(X,Φ) =

(

x =



x(i)



∈ Ω(X) :





∈ E(X) for some ρ > 0

)

where E = `

and



||x(i)||



Using the space `

(X) studied by (Maddox, 1988) i.e.

(X) =

(

x =



x(i)



∈ Ω(X) : x(i) ∈ X and

∞

∑

i=1

||x(i)||

= ||x(i)||

< ∞

)

we give the sufﬁcient and necessary condition a ma-

trix operator from the space `

(X) into the space

∃

(X,Φ) in the present paper.

2 MAIN RESULTS

Firstly, in this work we introduce the space E

∃

(X,Φ)

for E = `

, i.e.

∃

(X,Φ) =

(

x =



x(i)



∈ Ω(X ) :





∈ `

(X) for some ρ > 0

)

(

x =



x(i)



∈ Ω(X ) :



||x(i)||



∈ `

(X) for some ρ > 0

)

(

x =



x(i)



∈ Ω(X ) :

∞

∑

i=1



||x(i)||



< ∞ for some ρ > 0

)

Theorem 2.1. If ϕ

satisfy ∆

−condition for every

natural numbers i, then the following set `

∃

(X,Φ) be-

comes a linear space.

Proof.

(Oﬁe and Herawati, 2018).

Theorem 2.2. If ϕ

is a convex for every natural num-

bers i, then the space `

∃

(X,Φ) is a normed space with

the following norm.

||x||

= inf

(

ρ > 0 : Φ





≤ 1

)

Proof.

Obviously x = 0 in `

∃

(X,Φ) implies ||x||

= 0. Con-

versely, ||x||

= 0. Since Musielak-ϕ function, Φ, sat-

isﬁes convex property, we obtain

ϕ(x) = ϕ





≤



1/n



≤

for every n ∈ N. Therefore Φ(x) = 0, which implies

x = 0. The following step we will show the homoge-

neous property. It is clearly for real numbers α = 0,

then

Φ(αx) = 0.

Assume for α 6= 0. Since ||x||

≤ ρ, we have



αx

ρ|α|



≤ 1.

Thus ||αx||

≤ ρ|α|. Then ||αx||

≤ |α|||x||

, which

implies

||x||



|α|



≤

||αx||

Therefore |α|||x||

≤ ||αx||

for all real numbers α.

Thus, |α|||x||

= ||αx||

For next, we will show the triangle inequallity.

Since for every x, y ∈ `

∃

(X,Φ) and ϕ

satisfy convex

property for every i, for non-negative reall numbers

α,β with α +β = 1 we have

||x||

< α and ||y||

< β,

and



x + y

α + β



= Φ



α + β



≤

α + β





α + β





≤

α + β

= 1.

Consequently

||x + y||

≤ α + β

by deﬁnition of norm, then

||x + y|| ≤ ||x||

+ ||y||

Preposition 2.1. Let x ∈ `

∃

(X,Φ) for every

Musielak-ϕ function Φ = (ϕ

). If ||x||

≤ 1, then

Φ(x) ≤ ||x||

Proof.

Assume that for any vector x ∈ `

∃

(X,Φ), x 6= 0. By

the deﬁnition of the norm, we get

∑

i∈N



||x(i)||

||x||



≤ 1. (2.1)

ICOSTEERR 2018 - International Conference of Science, Technology, Engineering, Environmental and Ramiﬁcation Researches

980

For ||x||

= 1, we get

∑

i∈N



||x(i)||



≤ ||x||

Now, assume 0 < ||x||

< 1. Because ϕ

is an increas-

ing function for all i ∈ N and by ∆

-condition, from

(1) we conclude

∑

i∈N



||x(i)||



∑

i∈N



2||x(i)||

||x||



≤ ||x||

∑

i∈N



||x(i)||

||x||



≤ ||x||

By using this preposition, we proof this Theo-

rem as given below.

Theorem 2.3. Let A = (a

) be an inﬁnite real matrix.

Then A ∈ (`

(X),`

∃

(X,Φ)) if and only if for all x =

(x(i)) ∈ `

(X) there exists positive integer number m

such that

sup

||x||

≤1

∞

∑

n=1



||A

(x)||



≤ 1.

Proof.

(⇒) Using Zeller’s theorem, we get A ∈

(X),`

∃

(X,Φ)) is a continuous operator. Thus,

there exists a natural number m

that if

||x||

≤

implies

||Ax||

≤ 1 (2.2)

for all x = (x(i)) ∈ `

(X). By using preposition 2.2

we have

∞

∑

n=1





(x)





≤ 1.

Thus, we get

sup

||x||

≤1

∞

∑

n=1





(x)





= sup

||x||

≤1

∞

∑

n=1



||A

(x)||



≤ 1.

(⇐) Let A = (a

) an inﬁnite matrix and x = (x(i)) ∈

(X). We will show that Ax ∈ `

∃

(X,Φ). There exists

∈ N such that

sup

||x||

≤1

∞

∑

n=1



||A

(x)||



≤ 1.

Because for every x ∈ `

(X) with ||x||

≤

, we get

∞

∑

n=1



||A

(x)||



≤ 1 < ∞.

Since m

> 0, we have Ax ∈ `

∃

(X,Φ).

3 CONCLUSION

According to the main result, we conclude the suf-

ﬁcient and necessary condition for a matrix operator

acting from the space `

(X) into the space `

∃

(X,Φ).

ACKNOWLEDGEMENT

Special thanks to TALENTA USU 2018 for

funded and supported this paper.

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(Preprint)

Matrix Operator of Musielak-j Function Sequence Space

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