Faculty of Sciences and Mathematics, University of Niš, Serbia
Available at: http://www.pmf.ni.ac.yu/filomat
Filomat 22:2 (2008), 59–64_____________________________________________________
SOME NEW SEQUENCE SPACES AND ALMOST CONVERGENCE
Sameer Ahmad Gupkari
Abstract. The sequence space
acr
(a
r
c
have been defined and the classes
: lp)
and
(a
r
c
: c ) of
infinite matrices have been characterized by Aydin and Başar (On the new sequence spaces which
include the spaces
c0
and
c , Hokkaido Math. J. 33(2) (2004), 383-398) [1], where 1 ≤ p ≤ ∞ . The
main purpose of the present paper is to characterize the classes
f
and
(a
r
c
: f)
and
(a
r
c
: f 0 ) , where
f 0 denote the spaces of almost convergent and almost convergent null sequences with real or
complex terms.
2000 Mathematics Subject Classification: 46B20; 46E30; 40C05
Keywords and phrases. Sequence spaces of non absolute type, almost convergent sequences,
β − duals and matrix
mappings.
Received : July 12, 2008
Communicated by Vladimir Rakočević
1. Introduction
Let
ω be the space of all sequences, real or complex and
let l ∞ and c respectively
be the Banach spaces of bounded and convergent sequences x = ( xk ) with the usual
norm
x = sup xk . Let S : l ∞ → l ∞ be the shift operator defined by ( S x )n = xn + 1 for
k
all n∈
. A Banach limit L is defined on l ∞ , as a non negative linear functional such that
L ( Sx ) = L ( x ) and L(e) = 1 , e = (1,1,1 ,...) [2]. A sequence x ∈ l ∞ is said to be
almost convergent to the generalized limit α if all Banach limits of x are α [3]. We denote
the set of almost convergent sequences by f and almost convergent null sequences by f 0 ,
i.e.
{
={x∈l
}
( x ) = 0, uniformly in n}
f = x ∈ l ∞ : lim tm n ( x ) = α , uniformly in n
and
Where
and
f0
m
∞
: lim tm n
m
1 m
∑ xk + n ,
m +1 k =0
α = f − lim x .
tm n ( x ) =
t−1 , n = 0
60
Let
λ
μ be two sequence spaces and A = ( an k ) be an infinite matrix of real
and
= { 0 ,1, 2 ,... } . Then we say that A defines a
or complex numbers an k , where n , k ∈
λ
matrix mapping from
in to
μ , and denote it by writing A : λ → μ if for every sequence
x = ( xk )∈ λ , the sequence Ax =
{( Ax ) } , the A-transform of x, is in μ , where
n
( Ax )n = ∑ an k xk , ( n∈ )
(1.1)
k
For simplicity in notation, here and in what follows, the summation without limits runs from 0
to ∞ . We denote by ( λ : μ ) the class of all matrices A such that A : λ → μ . Thus
A ∈ ( λ : μ ) if and only if the series on the right side of (1.1) converges for every n ∈
and every x ∈ λ .
For a sequence space
λ , the matrix domain λA
of an infinite matrix A is defined by
λA = { x = ( xk ) ∈ ω : Ax ∈ λ}
The object of this paper is to characterize the classes
(a
r
c
: f ) and ( acr : f 0 ) of
infinite matrices.
r
r
The sequence space ac is defined as the set of all sequences whose A - transform
is in c [1], i.e.
⎧
⎫
1 n
1 + r k ) xk exists ⎬
acr = ⎨ x = ( xk ) ∈ ω : lim
(
∑
n →∞ n + 1
k =0
⎩
⎭
r
r
r
Where A denotes the matrix A = an k defined by
( )
anr k
⎧1 + r k
,
⎪
= ⎨ n +1
⎪0
,
⎩
(0 ≤ k ≤ n)
( k > n)
We refer the reader to [1] for relevant terminology and additional references on the
r
c
space a
.
2. Main Results
Define the sequence y = ( yk ( r ) ) , which will be used as the A - transform of a
r
sequence x = ( xk ) , i.e.
yk ( r ) =
k
∑
j =0
1+ r j
xj
k +1
; (k ∈
For brevity in notation, we write
60
)
(2.1)
61
a ( n , k , m) =
1
m +1
m
∑a
j =0
n+ j,k
and
⎡ a ( n , k , m) ⎤
⎡ a( n , k , m) a( n , k + 1, m) ⎤
−
a% (n , k , m ) = Δ ⎢
⎥ ( k + 1) = ⎢
⎥ ( k + 1)
k
k
1+ r k +1 ⎦
⎣ 1+ r
⎦
⎣ 1+ r
for n , k , m ∈ .
β
We denote by λ , the β -dual of a sequence space λ and mean the set of the
sequences x = ( xk ) such that x y = ( xk yk )∈ cs for all y = ( yk )∈ λ . Now, we may
give the following lemma which is needed in proving the Theorem (2.1) below.
r
r
Lemma 2.1[1]: Define the sets d1 and d 2 as follows
⎛ a ⎞
⎪⎧
⎪⎫
d1r = ⎨ a = ( ak ) ∈ ω : ∑ Δ ⎜ k k ⎟ (k + 1) < ∞ ⎬
k
⎝ 1+ r ⎠
⎪⎩
⎪⎭
⎧⎪
⎫⎪
⎛ a ⎞
and
d 2r = ⎨ a = ( ak ) ∈ ω : ⎜ k k ⎟ ∈ cs ⎬
⎪⎩
⎪⎭
⎝ 1+ r ⎠
ak + 1
⎛ ak ⎞
ak
where Δ ⎜
for all k ∈ .
=
−
k ⎟
k
1+ r
1+ r k + 1
⎝ 1+ r ⎠
Then
β
⎡⎣ acr ⎤⎦ = d1r ∩ d2r .
A ∈ ( acr : f
Theorem 2.1:
sup
m,n ∈
∑
) if and if
< ∞
a% (n , k , m)
(2.2)
k
⎧ an k ⎫
∈ cs for all n ∈ .
⎨
k ⎬
r
1
+
⎩
⎭k ∈
lim a% (n , k , m) = α k uniformly in n, for each k ∈
m →∞
lim
m →∞
Proof:
∑ a% (n , k , m) − α
k
= 0 uniformly in n.
(2.3)
(2.4)
(2.5)
k
Suppose that the conditions (2.2), (2.3), (2.4) and (2.5) hold and x ∈ ac . Then Ax
r
exists and at this stage, we observe from (2.4) and (2.2) that
k
∑α
j =0
j
≤
sup
m , n∈
∑
a% (n , j , m) < ∞
j
61
62
holds for every k ∈
. This gives that
(α k ) ∈ l1 . Since
x ∈ acr by the hypothesis, and
acr ≅ c , we have y ∈ c . Therefore, one can easily see that (α k yk ) ∈ l1 for each
y ∈ c and also there exists M > 0 such that sup yk < M . Now for any ε > 0 ,
k
choose a fixed k0 ∈
, there is some m0 ∈
k0
∑ [ a% (n , k , m) − α ] y
k
k= 0
ε
<
k
by (2.4) such that
2
for every m ≥ m0 , uniformly in n .
Also, by (2.5), there is some m1 ∈
∞
∑
, such that
ε
a% (n , k , m) − α k <
k = k0 + 1
2M
for every m ≥ m1 uniformly in n . Therefore, we have
1 m
∑ ( A x )n + i −
m +1 i = 0
∑α
k
k
∑ [ a% (n , k , m) − α k ] yk
∞0
∑ [ a% (n , k , m) − α ] y
+
k =0
<
<
ε
∞
∑
+
2
+ M
2
k
k = k0 + 1
k
a% (n , k , m) − α k yk
k = k0 + 1
ε
k
k
k0
≤
∑ [ a% (n , k , m) − α ] y
yk =
k
ε
= ε
2M
for all sufficiently large m, uniformly in n. Hence Ax ∈ f , which proves the sufficiency.
(
)
Conversely suppose that A ∈ acr : f . Then Ax exists for every x ∈ ac and
this implies that
Now
{a }
nk
β
k∈
∑ a ( n , k , m) x
k
∈ ⎡⎣acr ⎤⎦ for each n ∈
exists
for
each
m,
r
; the necessity of (2.3) is immediate.
n
and
x ∈ acr ,
the
sequence
k
am n = {a(n , k , m)}k ∈ define the continuous linear functionals φm n on acr by
φm n ( x) =
∑ a ( n , k , m) x
k
,
(m , n ∈ ) .
k
r
Since ac and c are norm isomorphic ([1], Theorem 2.2), it should follow with (2.1) that
φm n = a%m n
62
63
φm n
This just says that the functionals defined by
r
on ac are point wise bounded. Hence, by
the Banach- Steinhauss theorem, they are uniformly bounded, which yields that there exists a
constant M > 0 such that
φm n ≤ M
for all
m,n ∈
It therefore follows, using the complete identification just referred to that
∑
k
holds for all
To
b
(k )
a% (n , k , m) = φm n ≤ M
m,n ∈
prove
( r ) = {b ( r )}n ∈
(k )
n
(k )
n
b
which shows the necessity of the condition (2.2).
the
necessity
of
(2.4),
consider
the
∈ a for every k ∈
r
c
sequence
, where
n −k 1 + k
⎧
,
(k ≤ n ≤ k + 1)
⎪( −1)
1+ r k
(r ) = ⎨
⎪0
, ( 0 ≤ n ≤ k − 1 or
⎩
;
(n , k ∈ )
n > k +1 )
Since Ax exists and is in f for each x ∈ ac , one can easily see that
r
⎧⎪ ⎛ an k ⎞
⎫⎪
Ab( k ) (r ) = ⎨Δ ⎜
k
(
1)
+
∈ f
⎬
⎟
k
⎩⎪ ⎝ 1 + r ⎠
⎭⎪n ∈
for each k ∈ , which shows the necessity of (2.4).
r
Similarly by taking x = e ∈ ac , we also obtain that
⎧⎪
⎫⎪
⎛ an k ⎞
Ax = ⎨∑ Δ ⎜
k
(
1)
+
∈f
⎬
⎟
k
⎩⎪ k ⎝ 1 + r ⎠
⎭⎪n ∈
and this shows the necessity of (2.5) . This completes the proof.
If the space f is replaced by f 0 , then Theorem (2.1) is reduced to
Corollary 2.1:
αk = 0
A ∈ ( acr : f 0 ) if and if (2.2), (2.3) and (2.4), (2.5) also hold with
for all k ∈
.
References
[1] C. Aydin , Feyzi Başar, On the new sequence spaces which include the spaces c0
and c , Hokkaido Math. J. 33 (2)(2004), 383-398.
[2] S. Banach, Theòrie des operations linéaries, Warszawa, 1932.
[3] G. G. Lorentz, A contribution to the theory of divergent sequences, Acta Math.
63
64
80(1948), 167-190.
[4] S. Nanda, Matrix transformations and almost boundedness, Glasnik Mat. (34),
14(1979), 99-107.
Department of Mathematics, National Institute of Technology, Srinagar-190006, India
E-mail: [email protected]
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