$C_\lambda$-semiconservative $FK$-spaces
We study $C_\lambda$-semiconservative $FK$-spaces for $C_\lambda$-methods defined by deleting a set of rows from the Cesaro matrix $C_1$ and give some characterizations.
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UDC 517.95
I. Dağadur (Mersin Univ., Turkey)
Cλ-SEMICONSERVATIVE FK-SPACES
Cλ-НАПIВКОНСЕРВАТИВНI FK-ПРОСТОРИ
We study Cλ-semiconservative FK-spaces for Cλ-methods defined by deleting a set of rows from the Cesáro matrix C1
and give some characterizations.
Вивчено Cλ-напiвконсервативнi FK-простори для Cλ-методiв, що визначаються видаленням групи рядкiв iз матрицi
Чезаро C1, i наведено деякi характеристики.
1. Introduction and notation. The definition of semiconservative FK-space and some properties
of this space was given by Snyder and Wilansky in [14]. Ince, in [8], continued to work on Cesáro
semiconservative FK-space and to give some characterizations. In Section 2, for an FK-space X,
the concepts of Cλ-semiconservative FK-space have been defined. Their relationship to Cesáro semi-
conservative space and Cλ-semiconservative have also been examined. However, we study the Cλ-
semiconservative of the absolute summability domain lA, and show that if lA is Cλ-semiconservative,
then A cannot be l-replaceable. In Section 3 we study the subspaces CλF+, CλF, CλB and CλB+
of an FK-space X. In Section 4 we solve the problem of characterizing matrices A such that YA is
Cλ-semiconservative space for given Y.
Let F be an infinite subset of N and F as the range of a strictly increasing sequence of positive
integers, say F = {λ(n)}∞n=1 . The Cesáro submethod Cλ is defined as
(Cλx)n =
1
λ(n)
λ(n)∑
k=1
xk, n = 1, 2, . . . ,
where {xk} is a sequence of a real or complex numbers. Therefore, the Cλ-method yields a subse-
quence of the Cesáro method C1, and hence it is regular for any λ. Cλ is obtained by deleting a set
of rows from Cesáro matrix. The basic properties of Cλ-method can be found in [1] and [10].
Let s denote the space of all real or complex-valued sequences. It can be topologized with the
seminorms pn(x) = |xn| , n = 1, 2, . . . , and any vector subspace of s is called a sequence space.
A sequence space X, with a vector space topology τ, is a K-space provided that the inclusion
mapping i : (X, τ)→ s, i(x) = x is continuous. If, in addition, τ is complete, metrizable and locally
convex then (X, τ) is called an FK-space. So an FK-space is a complete, metrizable locally convex
topological vector space of sequences for which the coordinate functionals are continuous. The basic
properties of such spaces may be found in [1 – 13, 15].
By c0, l
∞ we denote the spaces of all number sequences that converge to zero and bounded
sequences, respectively. These are FK-spaces under ‖x‖ = supn |xn| .
As usual, l1 =
{
x ∈ s :
∑∞
n=1
|xn| <∞
}
is denoted simply by l. cs =
{
x ∈ s :
∑∞
n=1
xn
exists
}
, the space of all summable sequences; and bs is as the following:
bs =
{
x ∈ s : sup
k
∣∣∣∣∣
k∑
n=1
xn
∣∣∣∣∣ <∞
}
.
c© I. DAĞADUR, 2012
908 ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 7
Cλ-SEMICONSERVATIVE FK-SPACES 909
The sequence spaces
σs(λ) =
x ∈ s : lim
n
1
λ(n)
λ(n)∑
k=1
k∑
j=1
xj exists
,
σb(λ) =
x ∈ s : sup
n
∣∣∣∣∣∣ 1
λ(n)
λ(n)∑
k=1
k∑
j=1
xj
∣∣∣∣∣∣ <∞
and
q(λ) :=
x :
∞∑
j=1
λ(j)
∣∣42xj
∣∣ <∞ and x ∈ l∞
, q0(λ) := q(λ) ∩ c0
is FK-space with the norms [2, 3, 5 – 7]
‖x‖σb(λ) = sup
n
∣∣∣∣∣∣ 1
λ(n)
λ(n)∑
k=1
k∑
j=1
xj
∣∣∣∣∣∣ ,
‖x‖q(λ) =
∞∑
j=1
λ(j)
∣∣42xj
∣∣+ sup
n
|xj | ,
where
4xj = xj − xj+1 and 42xj = 4xj −4xj+1.
Throughout the paper e denotes the sequences of ones, (1, 1, . . . , 1, . . .); δj , j = 1, 2, . . . , the
sequence (0, 0, . . . , 0, 1, 0, . . .) with the one in the j th position; φ the linear span of the δj’s. The
topological dual of X is denoted by X ′. The space X is said to have AD if φ is dense in X. A
sequence x in a locally convex sequence space X is said the property AK (respectively σK(λ))
if x(n) → x
(
respectively
1
λ(n)
∑λ(n)
k=1
x(k) → x
)
in X where x(n) = (x1, x2, . . . , xn, 0, . . .) =
=
∑n
k=1
xkδ
k. An FK-space X is called Cesáro semiconservative space if Xf ⊂ σs where σs :=
:=
{
x ∈ s : limn
1
n
∑n
k=1
∑k
j=1
xj exists
}
(see [8]). Every AK space is a σK(λ). We recall (see
[5, 6, 13, 14]) that the f, β, σ, σb, σ(λ) and σb(λ)-dual of a subset X of s is defined to be
Xf =
{{
f(δk)
}
: f ∈ X ′
}
,
Xβ =
{
x ∈ s :
∞∑
k=1
xkyk exists for all y ∈ X
}
=
= {x ∈ s : xy = (xkyk) ∈ cs for all y ∈ X} ,
Xσ =
x ∈ s : lim
n
1
n
n∑
k=1
k∑
j=1
xjyj exists for all y ∈ X
=
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 7
910 I. DAĞADUR
= {x ∈ s : xy ∈ σs for all y ∈ X} ,
Xσb =
x ∈ s : sup
n
1
n
∣∣∣∣∣∣
n∑
k=1
k∑
j=1
yj
∣∣∣∣∣∣ <∞ for all y ∈ X
=
= {x ∈ s : xy ∈ σb for all y ∈ X} ,
Xσ(λ) =
x ∈ s : lim
n
1
λ(n)
λ(n)∑
k=1
k∑
j=1
xjyj exists for all y ∈ X
=
= {x ∈ s : xy ∈ σs(λ) for all y ∈ X} ,
Xσb(λ) =
x ∈ s : sup
n
1
λ(n)
∣∣∣∣∣∣
λ(n)∑
k=1
k∑
j=1
yj
∣∣∣∣∣∣ <∞ for all y ∈ X
=
= {x ∈ s : xy ∈ σb(λ) for all y ∈ X} ,
where xy = (xnyn). Let E, E1 be sets of sequences. Then for k = β, σ, σb, σ(λ) and σb(λ)
(a) E ⊂ Ekk,
(b) Ekkk = Ek,
(c) if E ⊂ E1 then Ek1 ⊂ Ek
holds. Also, if φ ⊂ E ⊂ E1 then Ef1 ⊂ Ef .
We shall be concerned with matrix transformations y = Ax, where x, y ∈ s, A = {ank}∞n,k=1 is
an infinite matrix with complex coefficients, and
yn =
∞∑
k=1
ankxk, n = 1, 2, . . . .
The sequence {ank}∞k=1 is called the n th row of A and is denoted by an, n = 1, 2, . . . ; similarly,
the k th column of the matrix A, {ank}∞n=1 is denoted by ak, k = 1, 2, . . . . For an FK-space Y, we
consider the summability domain YA defined by
YA = {x ∈ s : Ax exists and Ax ∈ Y } .
Then YA is an FK-space under the seminorms pn(x) = |xn|, n = 1, 2, . . . ;
hn(x) = sup
m
∣∣∣∣∣
m∑
k=1
ankxk
∣∣∣∣∣ , n = 1, 2, . . . , and (q ◦A)(x) = q(Ax) (see[13]).
2. Cλ-semiconservative FK-spaces. In this section, the concept of Cλ-semiconservative an
FK-space X containing φ is defined, and several theorems on this subject are given.
Definition 2.1. An FK-space X is called Cλ-semiconservative space if
Xf ⊂ σs(λ).
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 7
Cλ-SEMICONSERVATIVE FK-SPACES 911
This means that φ ⊂ X and
{
1
λ(n)
∑λ(n)
k=1
e(k)
}
is convergent for each f ∈ X ′.
For example, c0 is a Cλ-semiconservative FK-space. Every semiconservative FK-space is a Cλ-
semiconservative FK-space. But every Cλ-semiconservative FK-space is not a semiconservative FK-
space. An example of FK-space which is Cλ-semiconservative but not semiconservative is given in
[8] in case λ(n) = n.
The theorem below gives us the equivalence of Cesáro semiconservative and Cλ-semiconservative
of an FK-space X.
Theorem 2.1. Let X be an FK-space with φ ⊂ X and Xf ⊂ bs. Let λ := {λ(n)} be an
infinite subset of N such that lim supn
λ(n+ 1)
λ(n)
= 1. Then X is C1-semiconservative if and only if
it is Cλ-semiconservative.
Proof. Necessity is trivial.
Sufficiency. Let X be Cλ-semiconservative. Then for each f ∈ X ′, we have
lim
n
1
λ(n)
λ(n)∑
k=1
k∑
j=1
f
(
δj
)
exists.
Let tk (f) :=
∑k
j=1
f
(
δj
)
. So, (tk (f)) is Cλ-summable. Since Xf ⊂ bs, for all f ∈ X ′, (tk(f)) ∈
∈ l∞. Since lim supn
λ(n+ 1)
λ(n)
= 1, by Theorem 2.1 of [10], it is C1-summable. Therefore, X is a
C1-semiconservative space.
Using the same technique one can get the following theorem.
Theorem 2.2. Let X be an FK-space with φ ⊂ X, Xf ⊂ bs and λ := {λ(n)} , µ := {µ(n)}
infinite subsets of N. If limn
µ(n)
λ(n)
= 1, then X is Cλ-semiconservative if and only if it is Cµ-
semiconservative.
To see that limn
µ(n)
λ(n)
= 1 is not a necessary condition in Theorem 2.2, simply consider the
sequences λ(n) = n2 and µ(n) = n3. Then limn
λ(n+ 1)
λ(n)
= limn
µ(n+ 1)
µ(n)
= 1, and hence,
by Theorem 2.1, X is Cλ-semiconservative if and only if it is C1-semiconservative and X is Cµ-
semiconservative if and only if it is C1-semiconservative. However, limn
µ(n)
λ(n)
=
n3
n2
6= 1.
In Theorem 2.1, with lim supn
λ(n+ 1)
λ(n)
= 1 replaced by limn
λ(n+ 1)
λ(n)
= 1, the following
result is easily obtained by Theorem 2.2.
Corollary 2.1. Let limn
λ(n+ 1)
λ(n)
= 1. Then X is C1-semiconservative if and only if it is
Cλ-semiconservative.
The definition of a Cλ-conull FK-space X with φ ⊂ X, can be given by using Cλ-semi-
conservativity. A Cλ-semiconservative space X is called Cλ-conull, if
f (e) = lim
n
1
λ(n)
λ(n)∑
k=1
k∑
j=1
f
(
δj
)
,
for all f ∈ X ′. A Cλ-semiconservative space need not contain e but Cλ-conull must contain e.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 7
912 I. DAĞADUR
Theorem 2.3. If XA is a Cλ-conull FK-space, then it is a Cλ-semiconservative space.
Proof. Suppose that XA is Cλ-conull FK-space. Then
f (e) = lim
n
1
λ(n)
λ(n)∑
k=1
k∑
j=1
f
(
δj
)
,
for all f ∈ X ′A. Hence Xf
A ⊂ σs(λ).
We recall that, in [9] it is defined that a matrix A is l-replaceable if there is a matrix B = (bnk)
with lA = lB and
∑∞
n=1
bnk = 1 for all k ∈ N.
Theorem 2.4. If a matrix A is l-replaceable, then lA is not a Cλ-semiconservative FK-space.
Proof. If A is l-replaceable, then there is f ∈ l′A such that f
(
δj
)
= 1 for all j ∈ N in [9]. Hence
limn
1
λ(n)
∑λ(n)
k=1
∑k
j=1
f
(
δj
)
does not exist since
1
λ(n)
λ(n)∑
k=1
k∑
j=1
f
(
δj
)
=
λ(n) + 1
2
,
so lA is not Cλ-semiconservative space.
Theorem 2.5. (i) An FK-space that contains a Cλ-semiconservative FK-space must be a Cλ-
semiconservative FK-space.
(ii) A closed subspace, containing φ, of a Cλ-semiconservative FK-space is a Cλ-semiconservative
FK-space.
(iii) A countable intersection of Cλ-semiconservative FK-spaces is a Cλ-semiconservative FK-
spaces.
The proof is easily obtained from elementary properties of FK-spaces (see [13]).
Theorem 2.6. Let X be an FK-space containing φ. Then
(i) Xβ ⊂ Xσ(λ) ⊂ Xσb(λ) ⊂ Xf ,
(ii) if X is a σK(λ)-space, then Xf = Xσ(λ),
(iii) if X is an AD-space, then Xσ(λ) = Xσb(λ).
Proof. (ii) Let v ∈ Xσ(λ) and define f(x) = lim
n
1
λ(n)
∑λ(n)
k=1
∑k
j=1
vjxj for x ∈ X. Then
f ∈ X ′ by the Banach – Steinhaus theorem of [13]. Also
f (δq) = lim
n
1
λ(n)
(λ(n)− (q − 1))vq = vq, q < λ(n),
so v ∈ Xf . Thus Xσ(λ) ⊂ Xf .
Now we show that Xf ⊂ Xσ(λ). Let v ∈ Xf . Since X is a σK(λ)-space
f(x) = lim
n
1
λ(n)
λ(n)∑
k=1
k∑
j=1
xjf
(
δj
)
= lim
n
1
λ(n)
λ(n)∑
k=1
k∑
j=1
vjxj
for x ∈ X, then v ∈ Xσ(λ). This completes the proof of (ii).
(iii) Let v ∈ Xσb(λ) and define fn(x) =
1
λ(n)
∑λ(n)
k=1
∑k
j=1
vjxj for x ∈ X. Then {fn} is
pointwise bounded, hence equicontinuous by [13]. Since
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 7
Cλ-SEMICONSERVATIVE FK-SPACES 913
lim
n
fn (δq) = vq, q < λ(n),
then φ ⊂ {x : limn fn(x) exists}. Hence {x : limn fn(x) exists} is closed subspace of X by the
Convergence lemma [13]. Since X is an AD space, then X = {x : limn fn(x) exists} = φ and then
limn fn(x) exists for all x ∈ X. Thus v ∈ Xσ(λ). The opposite inclusion is trivial.
(i) φ ⊂ X by hypothesis. Since φ is AD-space, then
Xσb(λ) ⊂
(
φ
)σb(λ)
=
(
φ
)σ(λ) ⊂ (φ )f = Xf
by (iii) and [13].
Theorem 2.7. zσ(λ) is a Cλ-semiconservative space if and only if z ∈ σs(λ).
Proof. Let zσ(λ) be a Cλ-semiconservative space. Then
(
zσ(λ)
)f ⊂ σs(λ). Since zσ(λ) is a
σK(λ)-space by [4], we have
(
zσ(λ)
)f
=
(
zσ(λ)
)σ(λ)
by Theorem 2.6 (ii). So since
{z} ∈
(
zσ(λ)
)σ(λ) ⊂ σs(λ),
we get z ∈ σs(λ).
Now let z ∈ σs(λ). Then (σs (λ))σ(λ) ⊂ zσ(λ) and hence(
zσ(λ)
)σ(λ) ⊂ (σs(λ))σ(λ)σ(λ) = σs (λ) in [5].
Since zσ(λ) is a σK (λ)-space, then
(
zσ(λ)
)f
=
(
zσ(λ)
)σ(λ) ⊂ σs(λ).
It is clear that σs(λ) is not a Cλ-semiconservative space. Because σs (λ) = eσ(λ) and e /∈ σs(λ).
Now we get following theorem.
Theorem 2.8. The intersection of all Cλ-semiconservative FK-spaces is q0.
Proof. Let the set of all (C1-semiconservative) Cλ-semiconservative spaces be (Γ(C1)) Γ(Cλ).
Since every C1-semiconservative FK-space is Cλ-semiconservative space we get Γ(C1) ⊂ Γ(Cλ).
Also
∩{X : X ∈ Γ(C1)} ⊂ ∩{X : X ∈ Γ(Cλ)} .
On the other hand Theorem 6 of [8] the intersection of all C1-semiconservative spaces is q0. Hence
q0 ⊂ ∩{X : X ∈ Γ(Cλ)} . Therefore, by Theorem 5 of [8] we have
q0 ⊂ ∩{X : X ∈ Γ(Cλ)} ⊂ ∩{zσ : z ∈ σs} = σsσ = q.
Also ∩{X : X ∈ Γ(Cλ)} ⊂ c0, since c0 is a Cλ-semiconservative space so
∩{X : X ∈ Γ(Cλ)} ⊂ q ∩ c0 = q0,
where
q :=
x :
∞∑
j=1
j
∣∣42xj
∣∣ <∞ and x ∈ l∞
and q0 = q ∩ c0.
Theorem 2.8 is proved.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 7
914 I. DAĞADUR
3. A relationship between the distinguished subsets and Cλ-semiconservative FK-spaces. In
this section we shall now study the subspaces CλF, CλF+, CλB and CλB+ of an FK-space X.
Let X be an FK-space with φ ⊂ X. Then
CλW := CλW (X) =
x ∈ X :
1
λ(n)
λ(n)∑
k=1
x(k) → x (weakly) in X
=
=
x ∈ X : f(x) =
1
λ(n)
λ(n)∑
k=1
k∑
j=1
xjf
(
δj
)
for all f ∈ X ′
,
CλS := CλS(X) =
x ∈ X :
1
λ(n)
λ(n)∑
k=1
x(k) → x
=
= {x ∈ X : x has σK(λ) in X} ,
CλF
+ := CλF
+(X) =
x : lim
n
1
λ(n)
λ(n)∑
k=1
k∑
j=1
xjf
(
δj
)
exists for all f ∈ X ′
=
=
{
x : {xnf(δn)} ∈ σs(λ) for all f ∈ X ′
}
,
CλB
+ := CλB
+(X) =
x :
1
λ(n)
λ(n)∑
k=1
x(k)
is bounded in X
=
=
{
x : {xnf(δn)} ∈ σb(λ) for all f ∈ X ′
}
.
Also CλF = CλF
+ ∩X and CλB = CλB
+ ∩X.
We note that subspaces CλW and CλS are closely related to Cλ-conullity of the FK-space X
(see [4]).
The theorems below gives us some characterizations which are analogous to those given in [13]
(Chapter 10).
Theorem 3.1. Let X be an FK-space with φ ⊂ X, z ∈ s. Then z ∈ CλF
+ if and only
if z−1X = {x : zx ∈ X} is a Cλ-semiconservative FK-space, where zx = {xnzn}, in particular
e ∈ CλF+ if and only if X is Cλ-semiconservative FK-space.
Proof. Let f ∈
(
z−1X
)′
. Then f(x) = αx+ g (zx), α ∈ φ, g ∈ Y ′, by [13] and
f(δn) = αn + g (zδn) = αn + g (znδ
n) = αn + zng (δn).
Thus, since α ∈ φ ⊂ σs(λ) then {f(δn)} ∈ σs(λ) if and only if {zng (δn)} ∈ σs(λ), i.e., z ∈ CλF+.
An FK-space is called bounded convex Cλ-semiconservative space if it is a Cλ-semiconservative
space and includes q(λ).
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 7
Cλ-SEMICONSERVATIVE FK-SPACES 915
Theorem 3.2. Let X be an FK-space with φ ⊂ X, z ∈ s. Then z ∈ CλF if and only if z−1X is
bounded convex Cλ-semiconservative FK-space, in particular e ∈ CλF if and only if X is bounded
convex Cλ-semiconservative FK-space.
Proof. Let z ∈ CλF. Then z ∈ X so e ∈ z−1X and since z ∈ CλF
+, z−1X is a Cλ-
semiconservative FK-space by Theorem 3.1. Thus z−1X is a bounded convex Cλ-semiconservative
FK-space.
Let z−1X be a bounded convex Cλ-semiconservative FK-space. Then z−1X is Cλ-semiconser-
vative FK-space and e ∈ z−1X so z ∈ X. Thus since z ∈ CλF+ by Theorem 3.1 and z ∈ X, then
z ∈ CλF.
Theorem 3.3. Let X be an FK-space with φ ⊂ X, z ∈ s. Then z ∈ CλB
+ if and only if
q0(λ) ⊂ z−1X, in particular e ∈ CλB+ if and only if q0(λ) ⊂ X.
Proof. Let f ∈
(
z−1X
)′
. Then f(δn) = αn + zng (δn) by [13]. Hence, since α ∈ φ ⊂ σs(λ),
then z ∈ CλB+ if and only if {zng (δn)} ∈ σb(λ), i.e., z ∈ CλB+.
Theorem 3.4. Let X be an FK-space with φ ⊂ X, z ∈ s. Then z ∈ CλB if and only if
q(λ) ⊂ z−1X, in particular e ∈ CλB if and only if q(λ) ⊂ X.
Proof. Let z ∈ CλB. Then z ∈ X so e ∈ z−1X and z ∈ CλB
+. Thus z−1X ⊃ q(λ) by
Theorem 3.3.
Let z−1X ⊃ q(λ), then z−1X ⊃ q0(λ) and e ∈ z−1X. Thus, since z ∈ CλB+ by Theorem 3.3
and z ∈ X, then z ∈ CλB.
4. Matrix domains. In this section we give simple conditions for the subspaces CλB and CλF
in the FK-space YA, which is depend on the choice of the FK-space Y and the matrix A. Also,
we solve the problem of characterizing matrices A such that YA is Cλ-semiconservative space for
given Y.
The theorems below gives us some results which are analogous to those given in [13] (Chapters 9
and 12).
Theorem 4.1. Let Y be an FK-space and A be a matrix. Then YA is a Cλ-semiconservative
space if and only if the columns of A are in Y and
{
g(ak)
}
∈ σs(λ) for each g ∈ Y ′, where ak is
the k th column of A, akn = ank.
Proof. Necessity. The columns of A are inY since YA ⊃ φ by definition of Cλ -semiconservative
space. Given g ∈ Y ′, let f(x) = g(Ax) for x ∈ YA, so f ∈ Y ′A by [13] (Theorem 4.4.2). Then
f(δk) = g(ak) and the result follows since Y f
A ⊂ σs(λ).
Sufficiency. We first note that each row of A belongs to σs(λ) since in the hypothesis we may
take g = Pn, where Pn(x) = xn. This yields{
g(ak)
}
=
{
Pn(akn)
}
= {ank} ∈ σs(λ), k = 1, 2, 3, . . . .
Hence sA ⊃ (σs(λ))β .
Now let f ∈ Y ′A. Then by Theorem 4.4.2 of [13],
f(x) =
∞∑
k=1
αkxk + g(Ax) with g ∈ Y ′,
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 7
916 I. DAĞADUR
α ∈ sβA =
{
x :
∞∑
n=1
xnyn convergent for all y ∈ sA
}
⊂ (σs(λ))ββ = σs(λ), in [5].
Thus
f(δk) = αk + g(ak);
by the hypotesis and the fact that α ∈ σs (λ) we have
{
f(δk)
}
∈ σs (λ). Thus Y f
A ⊂ σs(λ) and YA
is Cλ-semiconservative space.
Theorem 4.2. If YA is Cλ-semiconservative space then AT ∈
(
Y β, σs(λ)
)
, where AT denotes
transpose of matrix A.
Proof. Since YA ⊃ q0 by Theorem 2.8 then A ∈ (q0, Y ) . Hence
AT ∈
(
Y β, qf0
)
=
(
Y β, σb
)
, where σb =
x ∈ w : sup
n
∣∣∣∣∣∣ 1n
n∑
k=1
k∑
j=1
xj
∣∣∣∣∣∣ <∞
by [13] (Theorem 8.3.8]. Let z ∈ Y β and define g ∈ Y ′ by g(y) = zy using the Banach – Steinhaus
theorem [13], where zy =
∑∞
k=1
zkyk. Let f(x) = g(Ax) so that f ∈ Y ′A by [13] (Theorem 4.4.2).
Hence
{
f(δk)
}
∈ σs(λ). But
f(δk) =
∞∑
n=1
znank =
(
AT z
)
k
so
(
AT z
)
∈ σs(λ).
Theorem 4.3. Let Y be an FK-space with AK. Then YA is Cλ-semiconservative space if and
only if the columns of A belong to Y and AT ∈
(
Y β, σs(λ)
)
.
Proof. Necessity is trivial by Theorem 4.2.
Sufficiency. Let g ∈ Y ′, zn = g (δn) . Then z ∈ Y f = Y β by [13] (Theorem 7.2.7), so(
AT z
)
∈ σs(λ). But
(
AT z
)
k
=
∞∑
n=1
znank = g
( ∞∑
n=1
ankδ
n
)
= g(ak)
since Y has AK. Hence we getg(ak) ∈ σs(λ). Then YA is Cλ-semiconservative space by Theo-
rem 4.1.
Definition 4.1. A matrix A is called Cλ-semiconservative if cA is Cλ-semiconservative space.
This definition is given because summability theory deals with spaces of the form cA and with
FK-spaces whose properties generalize those of such spaces. It would be nice if we can extend
theorems about conservative spaces to Cλ-semiconservative spaces.
Theorem 4.4. A is Cλ-semiconservative if and only if
(i) a has convergent columns, i.e., cA ⊃ φ,
(ii) a ∈ σs(λ), where a = {ak}, ak = limn ank,
(iii) AT ∈ (l, σs(λ)) .
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Cλ-SEMICONSERVATIVE FK-SPACES 917
Proof. Necessity. (i) is by Definition 4.1; to prove (ii) apply Theorem 4.1 with g := lim; (iii) is
by Theorem 4.2.
Sufficiency. Let g ∈ c′. Then g(y) = χ lim y +
∑∞
n=1
tnyn, t ∈ l by [13]. If we take y = Ax;
x = δk in here we obtained g
(
ak
)
= χ lim ank + (tA)k , where (tA)k =
∑∞
n=1
tnank. Since
g
(
ak
)
∈ σs(λ) from (ii) and (iii) then by Theorem 4.1. the result is obtained.
Theorem 4.5. The following are equialent for an FK-space X.
(i) If A ∈ (X, X) then XA is Cλ-semiconservative space.
(ii) X is Cλ-semiconservative space.
Proof. (i) implies (ii): Take A = I.
(ii) implies (i): If A ∈ (X, X) then X ⊂ XA, hence XA is Cλ-semiconservative space by
Theorem 2.5.
Theorem 4.6. Let z ∈ s, Y be an FK-space, and A be a matrix such that φ ⊂ YA i.e., the
columns of A belong to Y. Then the following propositions are equivalent in YA :
(i) z ∈ CλB+,
(ii)
{
1
λ(r)
∑λ(r)
p=1
Az(p)
}
is bounded in Y,
(iii) YAz ⊃ q0(λ) where the matrix Az is (ankzk) ,
(iv)
{
zkg(ak)
}
∈ σb(λ) for each g ∈ Y ′, where ak is kth column of A.
Proof. (i)⇔ (iii): z ∈ CλB+ if and only if z−1YA ⊃ q0(λ), where
z−1YA = {x : zx ∈ YA} , zx = {xnzn} ⇔ YAz ⊃ q0(λ)
by z−1YA = YAz and Theorem 3.3.
(iii)⇔ (iv): Since q0(λ) is AD space and by hypothesis then
Y f
Az ⊂ (q0(λ))f
by [13] (Theorem 8.6.1). Hence f(δk) = αk + g(aknzk) for each f ∈ Y ′Az with α ∈ sβAz, g ∈ Y ′ by
[13] (Theorem 4.4.2). Since
α ∈ sβAz ⊂ Y
β
Az ⊂ σb (λ)
then
{
f(δk)
}
∈ σb(λ)⇔
{
zkg(ak)
}
∈ σb (λ) for each g ∈ Y ′.
(ii)⇔ (iv): (iv) is true if and only ifg
1
λ(r)
λ(r)∑
p=1
Az(p)
is bounded for each g ∈ Y ′ by [13] (Theorem 8.0.2), where
g
1
λ(r)
λ(r)∑
p=1
Az(p)
= g
1
λ(r)
λ(r)∑
p=1
p∑
k=1
ankzk
=
1
λ(r)
λ(r)∑
p=1
p∑
k=1
zkg(akn).
Theorem 4.7. Assume that z ∈ s, (Y, q ) is an FK-space, and A is a matrix such that φ ⊂ YA
i.e., the columns of A belong to Y. Then the following propositions are equivalent in YA :
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 7
918 I. DAĞADUR
(i) z ∈ CλF+,
(ii)
{
1
λ(r)
∑λ(r)
p=1
Az(p)
}
is weakly Cauchy in Y, i.e.,
{
g
(
1
λ(r)
∑λ(r)
p=1
Az(p)
)}
is convergent
for each g ∈ Y ′,
(iii) YAz is Cλ-semiconservative space,
(iv)
{
zkg(ak)
}
∈ σs(λ) for each g ∈ Y ′.
Proof. (i) ⇔ (ii): z ∈ CλF
+ ⇔ z−1YA is Cλ-semiconservative space if and only if YAz is
Cλ-semiconservative space by Theorem 3.1.
(iii)⇔ (ii): Since the k th column of Az is zkak and by Theorem 4.1, this equivalent is trivial.
(iii)⇔ (iv): By Theorem 4.1, since the k th column of Az is zkak.
Theorem 4.8. Let Y be an FK-space such that weakly convergent sequences are convergent
in the FK-topology, let A be a row finite matrix with φ ⊂ YA. Then CλS = CλW = CλF = CλF
+
in YA.
Proof. If z ∈ CλF+,
{
1
λ(r)
λ(r)∑
p=1
Az(p)
}
is weakly Cauchy in Y by Theorem 4.7, hence Cauchy
[13] (Theorem 12.0.2), hence convergent say
1
λ(r)
∑λ(r)
p=1
Az(p) → y. However
1
λ(r)
∑λ(r)
p=1
Az(p) →
→ z in sA since this is a σK(λ) space because of sA is an AK space [13]. Thus
1
λ(r)
∑λ(r)
p=1
Az(p) → Az in s. But
1
λ(r)
∑λ(r)
p=1
Az(p) → y in s since Y is an FK-space hence
y = Az so z ∈ CλS by [4].
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Received 06.05.10,
after revision — 11.01.12
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 7
|
| id | umjimathkievua-article-2627 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:27:09Z |
| publishDate | 2012 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/41/8758e81da622f25a7e828e88fe206741.pdf |
| spelling | umjimathkievua-article-26272020-03-18T19:31:34Z $C_\lambda$-semiconservative $FK$-spaces $C_\lambda$-напiвконсервативнi $FK$-простори Dagadur, I. Дагадур, І. We study $C_\lambda$-semiconservative $FK$-spaces for $C_\lambda$-methods defined by deleting a set of rows from the Cesaro matrix $C_1$ and give some characterizations. Вивчено $C_\lambda$-напiвконсервативнi $FK$-простори для $C_\lambda$-методiв, що визначаються видаленням групи рядкiв iз матрицi Чезаро $C_1$, i наведено деякi характеристики. Institute of Mathematics, NAS of Ukraine 2012-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2627 Ukrains’kyi Matematychnyi Zhurnal; Vol. 64 No. 7 (2012); 908-918 Український математичний журнал; Том 64 № 7 (2012); 908-918 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2627/2013 https://umj.imath.kiev.ua/index.php/umj/article/view/2627/2014 Copyright (c) 2012 Dagadur I. |
| spellingShingle | Dagadur, I. Дагадур, І. $C_\lambda$-semiconservative $FK$-spaces |
| title | $C_\lambda$-semiconservative $FK$-spaces |
| title_alt | $C_\lambda$-напiвконсервативнi $FK$-простори |
| title_full | $C_\lambda$-semiconservative $FK$-spaces |
| title_fullStr | $C_\lambda$-semiconservative $FK$-spaces |
| title_full_unstemmed | $C_\lambda$-semiconservative $FK$-spaces |
| title_short | $C_\lambda$-semiconservative $FK$-spaces |
| title_sort | $c_\lambda$-semiconservative $fk$-spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2627 |
| work_keys_str_mv | AT dagaduri clambdasemiconservativefkspaces AT dagadurí clambdasemiconservativefkspaces AT dagaduri clambdanapivkonservativnifkprostori AT dagadurí clambdanapivkonservativnifkprostori |