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Matrix Operator of Musielak-ϕ Function Sequence Space
M. Ofie
1
and E. Herawati
1∗
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
i
} 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 `
1
(X) into the space `
∃
(X,Φ).
1 INTRODUCTION
Let X = (X,k · k
X
) 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 ∆
2
-condition. For a ϕ-function, ϕ,
satisfied ∆
2
-condition, the following space, denoted
by `
∃
(ϕ) is a generalization of Orlicz sequence space
(Kolk, 2015) i. e.
`
∃
ϕ
=
x = (x
i
) : x
i
∈ R and ϕ
x
ρ
∈ `
1
for some ρ > 0
=
x = (x
i
) : x
i
∈ R and
∞
∑
i=1
ϕ
x
i
ρ
< ∞
for some ρ > 0
.
respected to the following norm
||x||
ϕ
= inf
ρ > 0 : ϕ
x
ρ
≤ 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
i
) is a bounded sequence with
0 ≤ p
i
≤ 1 for all natural number i and two others se-
quence space into the Orlicz sequence space, `
M
, 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
nk
). i.e. an operator
A : X → Y with
Ax = (A
n
(x)) ∈ Y, for any x ∈ X
for every x ∈ X and for every natural number n
A
n
(x) =
∑
k≥1
a
nk
x
k
< ∞.
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
th
entry for every natural num-
ber k.
Let `
∃
(X,Φ) for X and Φ = {ϕ
i
} 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
Copyright
c
2020 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
979