LE MATEMATICHE
Vol. LXVIII (2013) – Fasc. I, pp. 33–51
doi: 10.4418/2013.68.1.4
LACUNARY SEQUENCE SPACES DEFINED BY
A MUSIELAK-ORLICZ FUNCTION
KULDIP RAJ - SUNIL K. SHARMA
In this paper we introduce lacunary sequence spaces defined by a
Musielak-Orlicz function M = (Mk ) and a sequence of modulus functions F = ( fk ). We also make an effort to study some topological properties and inclusion relations between these spaces.
1.
Introduction and Preliminaries
Let l∞ and c denote the Banach spaces of bounded and convergent sequences
x = (xk ) normed by kxk = supk |xk |, respectively. Let σ be a one-to-one mapping
of the set of positive integers into itself such that σ m (n) = σ (σ m−1 (n)), m =
1, 2, 3, · · · . A continuous linear functional ϕ on l∞ is said to be an invariant
mean or a σ −mean if and only if
1. ϕ(x) ≥ 0 when the sequence x = (xn ) has xn ≥ 0 for all n,
2. ϕ(e) = 1, where e = (1, 1, 1, · · · ) and
3. ϕ({xσ (n) }) = ϕ({xn }) for all x ∈ l∞ .
Entrato in redazione: 25 aprile 2012
AMS 2010 Subject Classification: 40A05, 40C05, 40D05.
Keywords: Lacunary sequence, Difference sequence, Modulus function, Musielak-Orlicz function.
34
KULDIP RAJ - SUNIL K. SHARMA
For certain kinds of mappings σ every invariant mean ϕ extends the limit functional on the space c, in the sense that ϕ(x) = lim x for all x ∈ c. The set of all
σ −convergent sequences will be denoted by Vσ . If x = (xn ), set T x = (T xn ) =
(xσ (n) ). It can be shown in [20] that
n
o
Vσ = x ∈ l∞ : lim tmn (x) = le uniformly in n, l = σ − lim x ,
m
(1)
where tmn (x) = (xn +T xn +...+T m xn )/(m+1). The special case of (1) in which
σ (n) = n + 1 was given by Lorentz [7]. Several authors including Schaefer [20],
Mursaleen [12], Savaş [19] and many others have studied invariant convergent
sequences.
A bounded sequence x = (xk ) is said to be strongly σ −convergent to a number l
if and only if (|xk − l|) ∈ Vσ with σ - limit zero (see[13]). By [Vσ ], we denote the
set of all strongly σ −convergent sequences. It is known that c ⊂ [Vσ ] ⊂ Vσ ⊂ l∞ .
By a lacunary sequence θ = (ir ), r = 0, 1, 2, · · · , where i0 = 0, we shall mean an
increasing sequence of non-negative integers hr = (ir − ir−1 ) → ∞ as r → ∞.
The intervals determined by θ are denoted by Ir = (ir−1 , ir ] and the ratio ir /ir−1
will be denoted by qr . The space of lacunary strongly convergent sequences Nθ
was defined by Freedman [4] as follows:
n
o
1
Nθ = x = (xk ) : lim
|xk − L| = 0 for some L .
∑
r→∞ hr
k∈Ir
In [6], Kızmaz defined the sequence spaces
n
o
Z(∆) = x = (xk ) : (∆xk ) ∈ Z for Z = `∞ , c and c0 ,
where ∆x = (∆xk ) = (xk − xk+1 ). Et and Çolak [3] generalized the difference
sequence spaces to the sequence spaces
n
o
Z(∆n ) = x = (xk ) : (∆n xk ) ∈ Z for Z = `∞ , c and c0 ,
where n ∈ N, ∆0 x = (xk ), ∆x = (xk − xk+1 ),
∆n x = (∆n xk ) = (∆n−1 xk − ∆n−1 xk+1 ).
The generalized difference sequence has the following binomial representation
n
n
∆ (xk ) =
v
∑ (−1)
v=0
n
v
xk+v .
LACUNARY SEQUENCE SPACES
35
An Orlicz function M : [0, ∞) → [0, ∞) is convex and continuous such that
M(0) = 0, M(x) > 0 for x > 0. Let w be the space of all real or complex sequences x = (xk ). Lindenstrauss and Tzafriri [8] used the idea of Orlicz function
to define the following sequence space,
n
`M = x ∈ w :
∞
∑M
k=1
|x | o
k
<∞
ρ
which is called as an Orlicz sequence space. The space `M is a Banach space
with the norm
o
n
|x | ∞
k
≤1 .
||x|| = inf ρ > 0 : ∑ M
ρ
k=1
It is shown in [8] that every Orlicz sequence space `M contains a subspace isomorphic to ` p (p ≥ 1). An Orlicz function M satisfies ∆2 -condition if and only if
for any constant L > 1 there exists a constant K(L) such that M(Lu) ≤ K(L)M(u)
for all values of u ≥ 0.
A sequence M = (Mk ) of Orlicz functions is called a Musielak-Orlicz function
(see [11], [14]). A sequence N = (Nk ) defined by
Nk (v) = sup{|v|u − Mk (u) : u ≥ 0}, k = 1, 2, · · ·
is called the complementary function of a Musielak-Orlicz function M. For
a given Musielak-Orlicz function M, the Musielak-Orlicz sequence space tM
and its subspace hM are defined as follows:
n
o
tM = x ∈ w : IM (cx) < ∞ for some c > 0 ,
n
o
hM = x ∈ w : IM (cx) < ∞ for all c > 0 ,
where IM is a convex modular defined by
∞
∑ Mk (xk ), x = (xk ) ∈ tM .
IM (x) =
k=1
We consider tM equipped with the Luxemburg norm
n
x
o
||x|| = inf k > 0 : IM
≤1
k
or equipped with the Orlicz norm
||x||0 = inf
n1
o
1 + IM (kx) : k > 0 .
k
36
KULDIP RAJ - SUNIL K. SHARMA
A Musielak-Orlicz function M = (Mk ) is said to satisfy ∆2 -condition if there
1
exist constants a, K > 0 and a sequence c = (ck )∞
k=1 ∈ `+ (the positive cone of
1
` ) such that the inequality
Mk (2u) ≤ KMk (u) + ck
holds for all k ∈ N and u ∈ R+ , whenever Mk (u) ≤ a.
Let X be a linear metric space. A function p : X → R is called paranorm, if
1. p(x) ≥ 0, for all x ∈ X,
2. p(−x) = p(x), for all x ∈ X,
3. p(x + y) ≤ p(x) + p(y), for all x, y ∈ X,
4. if (λn ) is a sequence of scalars with λn → λ as n → ∞ and (xn ) is a sequence of vectors with p(xn − x) → 0 as n → ∞, then p(λn xn − λ x) →
0 as n → ∞.
A paranorm p for which p(x) = 0 implies x = 0 is called total paranorm and the
pair (X, p) is called a total paranormed space. It is well known that the metric
of any linear metric space is given by some total paranorm (see [21], Theorem
10.4.2, P-183). For more detail about sequence spaces (see [2], [15], [16]) and
references therein.
A sequence space E is said to be solid or normal if (αk xk ) ∈ E whenever (xk ) ∈ E
and for all sequences of scalars(αk ) with |αk | ≤ 1 (see [14]).
The following inequality will be used throughout the paper. If 0 ≤ pk ≤ sup pk =
G, K = max(1, 2G−1 ) then
|ak + bk | pk ≤ K{|ak | pk + |bk | pk }
(2)
for all k and ak , bk ∈ C. Also |a| pk ≤ max(1, |a|G ) for all a ∈ C.
Let M = (Mk ) be a Musielak-Orlicz function and X be a locally convex Hausdorff topological linear space whose topology is determined by a set Q of seminorms q. Let p = (pk ) be a bounded sequence of positive real numbers and
u = (uk ) be a sequence of strictly positive real numbers. By w(X) we denotes
the space of all X-valued sequences. In this paper we define the following sequence spaces:
h
i
Vσ , ∆n , θ , M, u, p, q
1
n
h uk ∆n x k − l i pk
1
σ (m)
= x ∈ w(X) : lim
= 0,
∑ Mk q
r→∞ hr
ρ
k∈Ir
o
uniformly in m, for some ρ > 0 ,
37
LACUNARY SEQUENCE SPACES
h
i
n
h uk ∆n x k i pk
1
σ (m)
Vσ , ∆n , θ , M, u, p, q = x ∈ w(X) : lim
Mk q
∑
r→∞
hr k∈Ir
ρ
0
o
= 0, uniformly in m, for some ρ > 0
and
h
i
n
h uk ∆n x k i pk
1
σ (m)
Vσ , ∆n , θ , M, u, p, q = x ∈ w(X) : sup ∑ Mk q
ρ
∞
r,m hr k∈Ir
o
< ∞, for some ρ > 0 .
If we take M(x) = x, we get
h uk ∆n x k − l i pk
h
i
n
1
σ (m)
= 0,
Vσ , ∆n , θ , u, p, q = x ∈ w(X) : lim
∑ q
r→∞ hr
ρ
1
k∈Ir
o
uniformly in m, for some ρ > 0 ,
h
i
n
h uk ∆n x k i pk
1
σ (m)
Vσ , ∆n , θ , u, p, q = x ∈ w(X) : lim
= 0,
∑ q
r→∞ hr
ρ
0
k∈Ir
uniformly in m, for some ρ > 0
o
and
h
i
h uk ∆n x k i pk
n
1
σ (m)
Vσ , ∆n , θ , u, p, q = x ∈ w(X) : sup ∑ q
< ∞,
h
ρ
∞
r,m r k∈Ir
o
for some ρ > 0 .
If we take p = (pk ) = 1 for all k, we get
h uk ∆n x k − l i
h
i
n
1
σ (m)
Vσ , ∆n , θ , M, u, q = x ∈ w(X) : lim
∑ Mk q
r→∞ hr
ρ
1
k∈Ir
o
= 0, uniformly in m, for some ρ > 0 ,
i
n
h
h uk ∆n x k i
1
σ (m)
n
Vσ , ∆ , θ , M, u, q = x ∈ w(X) : lim
= 0,
∑ Mk q
r→∞ hr
ρ
0
k∈Ir
o
uniformly in m, for some ρ > 0
38
KULDIP RAJ - SUNIL K. SHARMA
and
h
i
n
h uk ∆n x k i
1
σ (m)
Vσ , ∆n , θ , M, u, q = x ∈ w(X) : sup ∑ Mk q
< ∞,
ρ
∞
r,m hr k∈Ir
o
for some ρ > 0 .
The main purpose of this paper is to introduce and study some lacunary
sequence spaces defined by a Musielak-Orlicz function. We examine
some
h
n
topological properties and inclusion relations between the spaces Vσ , ∆ , θ , M,
i
u, p, q in the second section. Third section devoted to the study of lacunary
Z
sequence spaces defined by a sequence of modulus functions. We also examine
some topological
properties and inclusion relation between the spaces
h
i
n
Vσ , ∆ , θ , F, p, q . Throughout the paper Z will denote any one of the notation
Z
0, 1 or ∞.
2.
Lacunary sequence spaces defined by a Musielak-Orlicz function
In this section of the paper we study very interesting properties like linearity,
paranorm
and some attractive
inclusion relations between the spaces
h
i
n
Vσ , ∆ , θ , M, u, p, q .
Z
Theorem 2.1. For any Musielak-Orlicz function M = (M
h k ) and for a bounded
i
sequence of positive real numbers p = (pk ), the spaces Vσ , ∆n , θ , M, u, p, q
Z
are linear over the field of complex numbers C.
h
Proof. Let x = (xk ), y = (yk ) ∈ Vσ , ∆n , θ , M, u, p, q]0 and let α, β ∈ C. Then
there exist positive numbers ρ1 and ρ2 such that
h uk ∆n x k i pk
1
σ (m)
=0
∑ Mk q
r→∞ hr
ρ
1
k∈Ir
lim
and
h uk ∆n y k i pk
1
σ (m)
lim
= 0.
∑ Mk q
r→∞ hr
ρ
2
k∈Ir
Define ρ3 = max(2|α|ρ1 , 2|β |ρ2 ). Since M = (Mk ) is non-decreasing and con-
39
LACUNARY SEQUENCE SPACES
vex, q is a seminorm and so by using inequality (2), we have
h αuk ∆n x k + β uk ∆n y k i pk
1
σ (m)
σ (m)
∑ Mk q
hr k∈I
ρ
3
r
h αuk ∆n x k β uk ∆n y k i pk
1
σ (m)
σ (m)
≤
Mk q
+q
∑
hr k∈Ir
ρ3
ρ3
n
h
i
h
uk ∆n y k i pk
pk
uk ∆ xσ k (m)
1
1
σ (m)
≤ K ∑ Mk q
+ K ∑ Mk q
hr k∈Ir
ρ1
hr k∈Ir
ρ2
→0
as r → ∞ uniformly in m.
h
i
This proves that Vσ , ∆n , θ , M, u, p, q is a linear space. Similarly, we can
h
i
h0
i
n
prove that Vσ , ∆ , θ , M, u, p, q and Vσ , ∆n , θ , M, u, p, q are linear spaces.
1
∞
Theorem 2.2. For any Musielak-Orlicz function M = (M
h k ) and p = (pk ) beia
bounded sequence of positive real numbers, the spaces Vσ , ∆n , θ , M, u, p, q
Z
are paranormed spaces, paranormed defined by
n pn h u ∆n x k (m) i pk H1
k
σ
g(x) = inf ρ H : sup Mk q
≤ 1, ρ > 0,
ρ
k≥1
o
uniformly in m ,
where H = max(1, supk pk ).
h
i
Proof. We shall prove the result for the case Vσ , ∆n , θ , M, u, p, q . Clearly
∞
g(x) =hg(−x) and g(θ ) = 0 iwhere θ is the zero sequence of X. Let x = (xk ), y =
(yk ) ∈ Vσ , ∆n , θ , M, u, p, q . Then there exist positive numbers ρ1 and ρ2 such
∞
that
h u ∆n x k (m) i pk
k
σ
sup Mk q
≤ 1, uniformly in m
ρ1
k≥1
and
h u ∆n y k (m) i pk
k
σ
sup Mk q
≤ 1, uniformly in m.
ρ
2
k≥1
40
KULDIP RAJ - SUNIL K. SHARMA
Let ρ = ρ1 + ρ2 and by using Minkowski’s inequality, we have
h u ∆n x k (m) + u ∆n y k (m) i pk
k
k
σ
σ
sup Mk q
ρ
k≥1
h u ∆n x k (m) + u ∆n y k (m) i pk
k
k
σ
σ
= sup Mk q
ρ1 + ρ2
k≥1
ρ
h u ∆n x k (m) i pk
1
k
σ
≤
sup Mk q
ρ1 + ρ2 k≥1
ρ1
ρ
h u ∆n y k (m) i pk
2
k
σ
+
sup Mk q
ρ1 + ρ2 k≥1
ρ2
≤ 1, uniformly in m.
Hence
g(x + y)
h u ∆n x k (m) + u ∆n y k (m) i pk H1
n
pn
k
k
σ
σ
≤ 1,
= inf (ρ1 + ρ2 ) H : sup Mk q
ρ
k≥1
o
ρ > 0, uniformly in m
n
h u ∆n x k (m) i pk H1
pn
k
σ
≤ 1,
≤ inf (ρ1 ) H : sup Mk q
ρ
1
k≥1
o
ρ1 > 0, uniformly in m
h u ∆n y k (m) i pk H1
n
pn
k
σ
+ inf (ρ2 ) H : sup Mk q
≤ 1,
ρ
2
k≥1
o
ρ2 > 0, uniformly in m
= g(x) + g(y).
Finally, we prove that the scalar multiplication is continuous. Let λ be any
complex number. By definition, we have
n pn h λ u ∆n x k (m) i pk H1
k
σ
H
g(λ x) = inf ρ : sup Mk q
≤ 1,
ρ
k≥1
o
ρ > 0, uniformly in m
n
h u ∆n x k (m) i pk H1
pn
k
σ
H
= inf (|λ |t) : sup Mk q
≤ 1, t > 0,
t
k≥1
o
uniformly in m ,
where t =
ρ
|λ | .
This completes the proof of the theorem.
41
LACUNARY SEQUENCE SPACES
00
00
0
0
funcTheorem 2.3. Let M
h = (Mk ) and M =i (Mkh) be two Musielak-Orlicz
i
tions. Then we have Vσ , , ∆n θ , M0 , u, p, q ∩ Vσ , ∆n , θ , M00 , u, p, q ⊆
Z
Z
h
i
Vσ , ∆n , θ , M0 + M00 , u, p, q .
Z
Proof. The proof is easy so we omit it.
Theorem 2.4. Let M = (Mk ) be a Musielak-Orlicz function, p = (pk ) be a
bounded sequence of positive real numbers and q1 , q2 are two seminorms on X.
Then
h
i h
i
Vσ , ∆n , θ , M, u, p, q1 ∩ Vσ , ∆n , θ , M, u, p, q2 6= ∅.
Z
Z
h
i
Proof. The zero elements belongs to Vσ , ∆n , θ , M, u, p, q1 and
Z
h
i
Vσ , ∆n , θ , M, u, p, q2 , thus the intersection is non empty.
Z
Theorem 2.5. For any Musielak-Orlicz function M = (Mk ), let q1 , q2 be two
seminorms on X. Then the following results holds:
(i) If q1 is stronger than q2 , then
h
i
h
i
Vσ , ∆n , θ , M0 , u, p, q1 ⊂ Vσ , θ , M0 , u, p, q2 ,
Z
Z
(ii)
h
i h
i
Vσ , ∆n , θ , M, u, p, q1 ∩ Vσ , ∆n , θ , M, u, p, q2
Z
hZ
i
⊂ Vσ , θ , M, u, p, q1 + q2 .
Z
Proof. The proof is easy so we omit it.
Theorem 2.6. Let M = (Mk ) be a Musielak-Orlicz function. Then
h
i
h
i
h
i
Vσ , ∆n , θ , M, u, p, q ⊂ Vσ , ∆n , θ , M, u, p, q ⊂ Vσ , ∆n , θ , M, u, p, q .
0
1
∞
h
i
h
i
Proof. The inclusion Vσ , ∆n , θ , M, u, p, q ⊂ Vσ , ∆n , θ , M, u, p, q is obvi1
h
i 0
n
ous. Let x = (xk ) ∈ Vσ , ∆ , θ , M, u, p, q . Then we have
1
h uk ∆n x k i pk
1
σ (m)
∑ Mk q
hr k∈I
ρ
r
h uk ∆n x k
i pk K
h l i pk
K
σ (m)−l
≤
M
q
+
k
∑
∑ Mk q ρ
hr k∈I
ρ
hr k∈I
r
r
nx
h
i
n h l iG o
p
u
∆
k
k
K
k
σ (m)−l
M
q
≤
+
K
max
1, Mk q
.
∑ k
hr k∈I
ρ
ρ
r
42
KULDIP RAJ - SUNIL K. SHARMA
h
i
Thus x = (xk ) ∈ Vσ , ∆n , θ , M, u, p, q . This completes the proof of the theo∞
rem.
Theorem 2.7. Let M = (Mk ) be a Musielak-Orlicz function and θ = (ir ) be a
lacunary sequence. Thenhthe following result
i holds:
h
i
n
(i) If lim inf qr > 1, then Vσ , ∆ , M, u, p, q ⊂ Vσ , ∆n , θ , M, u, p, q ,
r
Z
Z
h
i
h
i
n
0
n
0
(ii) If lim sup qr < ∞, then Vσ , ∆ , θ , M , u, p, q1 ⊂ Vσ , ∆ , M , u, p, q2 ,
Z
r
Z
(iii) If 1 < lim inf qr ≤ lim sup qr < ∞, then
r
r
h
i
h
i
Vσ , ∆n , M, u, p, q = Vσ , ∆n , θ , M, u, p, q .
Z
Z
Proof. The proof is easy so we omit it.
Theorem 2.8. Let 0 < pk ≤ tk and ptkk be bounded. Then
i
h
i
h
Vσ , ∆n , θ , M, u,t, q ⊂ Vσ , ∆n , θ , M, u, p, q .
Z
Z
h
Proof. We shall prove it for the case Z = 1. Let x = (xk ) ∈ Vσ , ∆n , θ , M, u,
i
t, q . We write
1
Sk =
h uk ∆n x k − l i pk
1
σ (m)
Mk q
∑
hr k∈Ir
ρ
and (µk ) = ptkk for all k ∈ N. Then 0 < µk ≤ 1 for all k ∈ N. Take 0 < µ < µk
for all k ∈ N. Define the sequence uk and vk as follows:
For Sk ≥ 1, let uk = Sk and vk = 0 and for Sk < 1, let uk = 0 and vk = Sk . Then
µ
µ
µ
clearly for all k ∈ N, we have Sk = uk + vk , Sk k = uk k + vk k . Now it follows that
µk
µk
µ
uk ≤ uk ≤ Sk and vk ≤ vk . Therefore,
1
1
1
1
µ
µ
µ
µ
Sk k =
(uk k + vk k ) ≤
Sk + ∑ vk .
∑
∑
∑
hr k∈Ir
hr k∈Ir
hr k∈Ir
hr k∈Ir
Now for each k,
1 µ 1 1−µ
1
µ
vk = ∑
vk
∑
hr k∈Ir
hr
k∈Ir hr
1 h 1 µ i µ1 µ h 1 1−µ i 1−µ
1−µ
≤ ∑
vk
∑
hr
hr
k∈Ir
k∈Ir
1
µ
=
∑ vk
hr k∈I
r
43
LACUNARY SEQUENCE SPACES
and so
1
µ
1
1
µ
Sk k ≤
Sk +
vk .
∑
∑
∑
hr k∈Ir
hr k∈Ir
hr k∈Ir
h
i
Hence x = (xk ) ∈ Vσ , ∆n , θ , M, u, p, q . Similarly we can prove other cases.
1
h
i
Theorem 2.9. The sequence space Vσ , ∆n , θ , M, u, p, q is solid.
∞
h
i
Proof. Let x = (xk ) ∈ Vσ , ∆n , θ , M, u, p, q , that is
∞
h uk ∆n x k i pk
1
σ (m)
< ∞.
∑ Mk q
hr k∈I
ρ
r
Let (αk ) be a sequence of scalars such that |αk | ≤ 1 for all k ∈ N. Thus we have
h αk uk ∆n x k i pk
1
σ (m)
∑ Mk q
hr k∈I
ρ
r
h uk ∆n x k i pk
1
σ (m)
∑ Mk q
hr k∈I
ρ
r
< ∞.
h
i
This shows that (αk xk ) ∈ Vσ , ∆n , θ , M, u, p, q for all sequences of scalars
h∞
i
(αk ) with |αk | ≤ 1 for all k ∈ N, whenever (xk ) ∈ Vσ , ∆n , θ , M, u, p, q . Hence
∞
h
i
n
the space Vσ , ∆ , θ , M, u, p, q is a solid sequence space. This completes the
∞
proof of the theorem.
h
i
Corollary 2.10. The sequence space Vσ , ∆n , θ , M, u, p, q is monotone.
≤
∞
Proof. It is easy to prove so we omit the details.
3.
Lacunary sequence spaces defined by a sequence of modulus functions
A modulus function is a function f : [0, ∞) → [0, ∞) such that
1. f (x) = 0 if and only if x = 0,
2. f (x + y) ≤ f (x) + f (y) for all x ≥ 0, y ≥ 0,
3. f is increasing
4. f is continuous from right at 0.
44
KULDIP RAJ - SUNIL K. SHARMA
It follows that f must be continuous everywhere on [0, ∞). The modulus funcx
tion may be bounded or unbounded. For example, if we take f (x) = x+1
, then
p
f (x) is bounded. If f (x) = x , 0 < p < 1, then the modulus f (x) is unbounded.
Subsequentially, modulus function has been discussed in ([1], [9], [10], [17],
[18]) and references therein.
Let F = ( fk ) be a sequence of modulus function, X be a locally convex Hausdorff topological linear space whose topology is determined by a set Q of seminorms q. Let p = (pk ) be a bounded sequence of positive real numbers and
u = (uk ) be a sequence of strictly positive real numbers. By w(X) be denote
the space of all X-valued sequences. In this section we define the following
sequence spaces:
h
i n
h ∆n x k − l i pk
1
σ (m)
n
Vσ , ∆ , θ , F, p, q = x = (xk ) ∈ w(X) : lim
fk q
∑
r→∞
hr k∈Ir
ρ
1
o
= 0, uniformly in m, for some ρ > 0 ,
h
i
n
h ∆n x k i pk
1
σ (m)
Vσ , ∆n , θ , F, p, q = x = (xk ) ∈ w(X) : lim
∑ fk q ρ
r→∞ hr
0
k∈Ir
o
= 0, uniformly in m, for some ρ > 0
and
h ∆n x k i pk
h
i
n
1
σ (m)
Vσ , ∆n , θ , F, p, q = x = (xk ) ∈ w(X) : sup ∑ fk q
ρ
∞
r,m hr k∈Ir
o
< ∞, for some ρ > 0 .
If we take F(x) = x, we get
h
i
n
h ∆n x k − l i pk
1
σ (m)
Vσ , ∆n , θ , p, q = x = (xk ) ∈ w(X) : lim
q
=0
∑
r→∞
hr k∈Ir
ρ
1
o
uniformly in m, for some ρ > 0 ,
h
i
n
h ∆n x k i pk
1
σ (m)
n
Vσ , ∆ , θ , p, q = x = (xk ) ∈ w(X) : lim
= 0,
∑ q ρ
r→∞ hr
0
k∈Ir
uniformly in m, for some ρ > 0
o
45
LACUNARY SEQUENCE SPACES
and
h
i
n
h ∆n x k i pk
1
σ (m)
Vσ , ∆n , θ , p, q = x = (xk ) ∈ w(X) : sup ∑ q
< ∞,
h
ρ
∞
r,m r k∈Ir
o
for some ρ > 0 .
If we take p = (pk ) = 1 for all k, we get
h ∆n x k − l i
h
i
n
1
σ (m)
n
fk q
Vσ , ∆ , θ , F, q = x = (xk ) ∈ w(X) : lim
∑
r→∞
hr k∈Ir
ρ
1
o
= 0, uniformly in m, for some ρ > 0 ,
h
i
n
h ∆n x k i
1
σ (m)
Vσ , ∆n , θ , F, q = x = (xk ) ∈ w(X) : lim
= 0,
fk q
∑
r→∞ hr
ρ
0
k∈Ir
o
uniformly in m, for some ρ > 0
and
h
i
n
h ∆n x k i
1
σ (m)
Vσ , ∆n , θ , F, q = x = (xk ) ∈ w(X) : sup ∑ fk q
< ∞,
h
ρ
∞
r,m r k∈Ir
o
for some ρ > 0 .
The main purpose of this section is to study some
topological iproperties and
h
some inclusion relations between of the spaces Vσ , ∆n , θ , F, p, q .
Z
Theorem 3.1. Let F = ( fk ) be a sequence of modulus functions and
h p = (pk )
be a bounded sequence of positive real numbers. Then the spaces Vσ , ∆n , θ , F,
i
p, q , Z = 0, 1, ∞ are linear over the field of complex numbers C.
Z
h
i
Proof. We shall prove the result for the case Vσ , ∆n , θ , F, p, q . Let x = (xk ),
0
h
i
n
y = (yk ) ∈ Vσ , ∆ , θ , F, p, q and let α, β ∈ C. Then there exist positive num0
bers ρ1 and ρ2 such that
h ∆n x k i pk
1
σ (m)
fk q
= 0, uniformly in m
∑
r→∞ hr
ρ1
k∈Ir
lim
46
KULDIP RAJ - SUNIL K. SHARMA
and
h ∆n y k i pk
1
σ (m)
fk q
= 0, uniformly in m.
∑
r→∞ hr
ρ2
k∈Ir
lim
Define ρ3 = max(2|α|ρ1 , 2|β |ρ2 ). Since F = ( fk ) is non-decreasing, q is a seminorm and so by using inequality (2), we have
h α∆n x k + β ∆n y k i pk
1
σ (m)
σ (m)
fk q
∑
hr k∈Ir
ρ3
h α∆n x k
β ∆n yσ k (m) i pk
1
σ (m)
) + q(
≤
∑ fk q
hr k∈I
ρ3
ρ3
r
h ∆n x k i pk
h ∆n y k i pk
1
1
σ (m)
σ (m)
≤ K ∑ fk q
+ K ∑ fk q
→0
hr k∈Ir
ρ1
hr k∈Ir
ρ2
as r → ∞ uniformly in m.
h
i
This proves that Vσ , ∆n , θ , F, p, q is a linear space. Similarly, we can prove
0
h
i
h
i
that Vσ , ∆n , θ , F, p, q and Vσ , ∆n , θ , F, p, q are linear spaces.
1
∞
Theorem 3.2. Let F = ( fk ) be a sequence of modulus functions and
h p = (pk ) be
a bounded sequence of positive real numbers. Then the spaces Vσ , ∆n , θ , F, p,
i
q are paranormed spaces, paranormed defined by
Z
∆n x k (m) pk i H1
o
n pn h
σ
≤ 1, ρ > 0 uniformly in m ,
g∗ (x) = inf ρ H : sup fk q
ρ
k≥1
where H = max(1, supk pk ).
h
i
Proof. We shall prove the theorem for the case Vσ , ∆n , θ , F, p, q . Clearly,
∞
∗
g∗ (x) = g(−x) and
θ is the zero sequence of X. Let x =
h g (θ ) = 0 where
i
n
(xk ), y = (yk ) ∈ Vσ , ∆ , θ , F, p, q . Then there exist positive numbers ρ1 and
∞
ρ2 such that
∆n x k (m) pk
σ
sup fk q
≤ 1, uniformly in m
ρ1
k≥1
and
∆n y k (m) pk
σ
sup fk q
≤ 1, uniformly in m.
ρ
2
k≥1
47
LACUNARY SEQUENCE SPACES
Let ρ = ρ1 + ρ2 and by using Minkowski’s inequality, we have
∆n x k (m) + ∆n y k (m) pk
∆n x k (m) + ∆n y k (m) i pk
σ
σ
σ
σ
sup fk q
= sup fk q
ρ
ρ
+
ρ
1
2
k≥1
k≥1
ρ
h ∆n x k (m) i pk ρ
h ∆n y k (m) i pk
1
2
σ
σ
≤
sup fk q
+
sup fk q
ρ1 + ρ2 k≥1
ρ1
ρ1 + ρ2 k≥1
ρ2
≤ 1, uniformly in m.
Hence
n
∆n x k (m) + ∆n y k (m) pk H1
pn
σ
σ
g∗ (x + y) = inf (ρ1 + ρ2 ) H : sup fk q
ρ
k≥1
o
≤ 1, ρ > 0, uniformly in m
n
∆n x k (m) pk H1
pn
σ
≤ inf (ρ1 ) H : sup fk q
≤ 1, ρ1 > 0,
ρ1
k≥1
o
uniformly in m
n
∆n y k (m) pk H1
pn
σ
+ inf (ρ2 ) H : sup fk q
≤ 1, ρ2 > 0,
ρ2
k≥1
o
uniformly in m
= g∗ (x) + g∗ (y).
Finally, we prove that the scalar multiplication is continuous. Let λ be any
complex number. By definition, we have
n pn λ ∆n x k (m) pk H1
σ
≤ 1, ρ > 0,
g∗ (λ x) = inf ρ H : sup fk q
ρ
k≥1
o
uniformly in m
n
∆n x k (m) pk H1
pn
σ
= inf (|λ |t) H : sup fk q
≤ 1,t > 0,
t
k≥1
o
uniformly in m ,
where t =
ρ
|λ | .
This completes the proof of the theorem.
Theorem 3.3. Let F 0 = ( fk0 ) and F 00 = ( fk00 ) be two sequences of modulus functions. Then we have
h
i h
i
h
i
Vσ , ∆n , θ , F 0 , p, q ∩ Vσ , ∆n , θ , F 00 , u, p, q ⊆ Vσ , ∆n , θ , F 0 + F 00 , u, p, q .
Z
Z
Z
48
KULDIP RAJ - SUNIL K. SHARMA
Proof. The proof is easy so we omit it.
Theorem 3.4. Let F = (Fk ) be a sequence of modulus functions and p = (pk )
be a bounded sequence ofh positive real numbers.
i h Then for any two
i seminorms
n
n
q1 and q2 on X, we have Vσ , ∆ , θ , F, p, q1 ∩ Vσ , ∆ , θ , F, p, q2 6= φ .
Z
Z
i
h
Proof. Since the zero element belongs to Vσ , ∆n , θ , F, p, q1 and
Z
h
i
Vσ , ∆n , θ , F, p, q2 and thus the intersection is non empty.
Z
Theorem 3.5. Let F = (Fk ) be a sequence of modulus functions. Then
h
i
i
i
Vσ , ∆n , θ , F, p, q ⊂ [Vσ , ∆n , θ , F, p, q ⊂ [Vσ , ∆n , θ , F, p, q .
0
1
∞
h
i
h
i
Proof. The inclusion Vσ , ∆n , θ , F, p, q ⊂ Vσ , ∆n , θ , F, p, q is obvious.
0
1
h
i
Let x = (xk ) ∈ Vσ , ∆n , θ , F, p, q . Then we have
1
h ∆n x k i pk
σ (m)
1
hr ∑ f k q
ρ
k∈Ir
h ∆n x k
i pk
h l i pk
1
1
σ (m)−l
f
q
+
K
k
∑
∑ fk q ρ
hr k∈I
ρ
hr k∈I
r
r
nx
h
i
n
h l iG o
p
∆
k
k
K
σ (m)−l
f
q
.
≤
+
K
max
1,
fk q
∑ k
hr k∈I
ρ
ρ
r
≤K
h
i
Thus x = (xk ) ∈ Vσ , ∆n , θ , F, p, q . This completes the proof of the theorem.
∞
Theorem 3.6. Let F = ( fk ) be a sequence of modulus functions and let θ = (ir )
be a lacunary sequence. Then
the following
holds:
h
i result
h
i
n
(i) If lim inf qr > 1. Then Vσ , ∆ , F, p, q ⊂ Vσ , ∆n , θ , F, p, q ,
r
Z i
Zi
h
h
(ii) if lim sup qr < ∞. Then Vσ , ∆n , θ , F, p, q ⊂ Vσ , ∆n , F, p, q ,
Z
r
Z
(iii) if 1 < lim inf qr ≤ lim sup qr < ∞. Then
r
r
h
i
h
i
Vσ , ∆n , F, p, q = Vσ , ∆n , θ , F, p, q .
Z
Proof. It is easy to prove so we omit it.
Z
49
LACUNARY SEQUENCE SPACES
Theorem 3.7. Let 0 < pk ≤ tk and ( ptkk ) be bounded. Then
h
i
h
i
Vσ , ∆n , θ , F,t, q ⊂ Vσ , ∆n , θ , F, p, q .
Z
Z
h
i
Proof. We will prove it for the case Z = 1. Let x = (xk ) ∈ Vσ , ∆n , θ , F,t, q .
1
We write
n
h ∆ x k i pk
1
σ (m)
Sk =
fk q
∑
hr k∈Ir
ρ
and (µk ) = ptkk for all k ∈ N. Then 0 < µk ≤ 1 for all k ∈ N. Take 0 < µ < µk
for all k ∈ N. Define the sequence uk and vk as follows:
For Sk ≥ 1, let uk = Sk and vk = 0 and for Sk < 1, let uk = 0 and vk = Sk . Then
µ
µ
µ
clearly for all k ∈ N, we have Sk = uk + vk , Sk k = uk k + vk k . Now it follows that
µ
µ
µ
uk k ≤ uk ≤ Sk and vk k ≤ vk . Therefore
1
1
1
1
µ
µ
µ
µ
Sk k =
(uk k + vk k ) ≤
Sk + ∑ vk .
∑
∑
∑
hr k∈Ir
hr k∈Ir
hr k∈Ir
hr k∈Ir
Now for each k,
1 µ 1 1−µ
1
µ
v
=
∑ k ∑ hr vk hr
hr k∈I
k∈Ir
r
1 h 1 µ i µ1 µ h 1 1−µ i 1−µ
1−µ
≤ ∑
vk
∑
hr
hr
k∈Ir
k∈Ir
1
µ
=
∑ vk
hr k∈I
r
and so
1
µ
1
1
µk
S
≤
S
+
v
∑ k hr ∑ k hr ∑ k .
hr k∈I
k∈Ir
k∈Ir
r
h
i
Hence x = (xk ) ∈ Vσ , ∆n , θ , F, p, q . Similarly we can prove for other cases.
1
h
i
Theorem 3.8. The sequence space Vσ , ∆n , θ , F, p, q is solid.
∞
h
i
Proof. Let x = (xk ) ∈ Vσ , ∆n , θ , F, p, q . Then
∞
h ∆n x k i pk
1
σ (m)
fk q
< ∞.
∑
hr k∈Ir
ρ
50
KULDIP RAJ - SUNIL K. SHARMA
Let (αk ) be a sequence of scalars such that |αk | ≤ 1 for all k ∈ N. Thus we have
h αk ∆n x k i pk
1
σ (m)
∑ fk q
hr k∈I
ρ
r
h ∆n x k i pk
1
σ (m)
∑ fk q ρ
hr k∈I
r
< ∞.
h
i
This shows that (αk xk ) ∈ Vσ , ∆n , θ , F, p, q for all sequences of scalars (αk )
∞ h
i
with |αk | ≤ 1 for all k ∈ N, whenever (xk ) ∈ Vσ , ∆n , θ , F, p, q . Hence the
∞
h
i
n
space Vσ , ∆ , θ , F, p, q is a solid sequence space. This completes the proof
∞
of the theorem.
h
i
Corollary 3.9. The sequence space Vσ , ∆n , θ , F, p, q is monotone.
≤
∞
Proof. It is easy to prove so we omit the details.
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KULDIP RAJ
School of Mathematics
Shri Mata Vaishno Devi University,
Katra-182320, J & K, India.
e-mail: [email protected]
SUNIL K. SHARMA
School of Mathematics
Shri Mata Vaishno Devi University,
Katra-182320, J & K, India.
e-mail: [email protected]