International Journal of Advanced and Applied Sciences, 4(4) 2017, Pages: 43-48
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International Journal of Advanced and Applied Sciences
Journal homepage: http://www.science-gate.com/IJAAS.html
43
On some new i-convergent double sequence spaces defined by a compact
operator
Vakeel A. Khan
1
, Hira Fatima
1
, Ayhan Esi
2,
*, Sameera A.A. Abdullah
1
, Kamal M.A.S. Alshlool
1
1
Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
2
Department of Mathematics, Adiyaman University, Adiyaman, Turkey
ARTICLE INFO ABSTRACT
Article history:
Received 29 December 2016
Received in revised form
29 February 2017
Accepted 3 March 2017
In this article we introduce and study some I-convergent double sequence
spaces 2
( ) ,
2
0
( ) ,
2
∞
( ) with the help of compact operator T
on the real space ℝ and an Orlicz function M. We study some of its
topological and algebraic properties and prove some inclusion relations on
these spaces.
Keywords:
Compact operator
Orlicz function
I- convergence
© 2017 The Authors. Published by IASE. This is an open access article under the CC
BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
*Let ℕ, ℝ, ℂ be the sets of all natural, real, and
complex numbers respectively. We denote
2
= { = (
)∶
∈ ℝ ℂ }
(1.1)
showing the space of all real or complex double
sequences.
Definition 1.1: Let X and Y be two normed linear
spaces. An operator ∶ → is said to be a
compact linear operator (or completely continuous
linear operator), if :(i) T is linear. (ii) T maps every
bounded sequence (
) in X onto a sequence T(
) in
Y which has a convergent subsequence.
The set of all compact linear operators (X, Y) is a
closed subspace of ℬ(X, Y) and (X, Y) is a Banach
space if Y is a Banach space. Throughout the paper,
we denote
2
∞
,
2
and
2
0
as the Banach spaces of
bounded, convergent, and null double sequences of
reals respectively with the norm:
║║ = sup
,∈ℕ
. (1.2)
Following Başar and Altay (2003) and Sengonul
(2007), we introduce the double sequence spaces
2
and
2
0
with the help of compact operator T on ℝ as
follows:
* Corresponding Author.
Email Address: aesi23@hotmail.com (A. Esi)
https://doi.org/10.21833/ijaas.2017.04.007
2313-626X/© 2017 The Authors. Published by IASE.
This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/)
2
= { = (
) ∈
2
∞
∶
() ∈
2
}
2
0
= { = (
) ∈
2
∞
∶
() ∈
2
0
}.
As a generalization of usual convergence, the
concept of statistical convergent was first introduced
by Fast (1951) and also independently by Buck
(1953) and Schoenberg (1959) for real and complex
sequences. Later on, it was further investigated from
a sequence space point of view and linked with the
Summability theory by Šalát (1980) and Tripathy
(2004).
Definition 1.2: A double sequence = (
)∈
2
is
said to be I-convergent to a number L if for every >
0, we have
{(, ) ∈ ℕ × ℕ ∶ |
− | ≥ } ∈ . (1.3)
In this case, we written − lim
(
)=. The
notation of ideal convergence (I-convergence) was
introduced and studied by Kostyrko et al. (2000,
2005). Later on, it was studied by Šalát et al. (2004,
2005), Tripathy and Hazarika (2009, 2011), Khan
and Ebadullah (2011, 2012, and 2013). Now, we
recall the following definitions:
Definition 1.3: Let X be a non-empty set. Then, a
family of sets I ⊆ 2
X
is said to be an Ideal in X if
1. φ ∈ I;
2. I is additive; that is, , ∈ ⇒ ∪ I ∈ ;
3. I is hereditary that is, ∈ , ⊆ ⇒ ∈ .
An Ideal ⊆ 2
is called non trivial if ≠ 2
.
A non-trivial ideal ⊆ 2
is called admissible if
{{} ∶ ∈ } ⊆ .