International Journal of Analysis and Applications

Volume 18, Number 4 (2020), 663-671

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-18-2020-663

INVARIANT SUMMABILITY AND UNCONDITIONALLY CAUCHY SERIES

NIMET PANCAROḠLU AKIN∗

Afyon Kocatepe University Department of Mathematics and Science Education

∗Corresponding author: npancaroglu@aku.edu.tr

Abstract. In this study, we will give new characterizations of weakly unconditionally Cauchy series and

unconditionally convergent series through summability obtained by the invariant convergence.

1. Introduction

Let σ be a mapping of the positive integers into itself. A continuous linear functional ϕ on m, the space

of real bounded sequences, is said to be an invariant mean or a σ mean, if and only if,

(1) φ(x) ≥ 0, when the sequence x = (xj) is such that xj ≥ 0 for all j,

(2) φ(e) = 1,where e = (1, 1, 1....),

(3) φ(xσ(j)) = φ(x) for all x ∈ m.

The mappings φ are assumed to be one-to-one and such that σi(j) 6= j for all positive integers j and i,

where σi(j) denotes the ith iterate of the mapping σ at j. Thus φ extends the limit functional on c, the

space of convergent sequences, in the sense that φ(x) = lim x for all x ∈ c. In case σ is translation mappings

σ(j) = j + 1, the σ mean is often called a Banach limit and Vσ, the set of bounded sequences all of whose

invariant means are equal, is the set of almost convergent sequences.

Received March 27th, 2020; accepted May 14th, 2020; published June 1st, 2020.

2010 Mathematics Subject Classification. 40A05.

Key words and phrases. unconditionally Cauchy series; invariant convergence; invariant convergent series.

©2020 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

663

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-18-2020-663


Int. J. Anal. Appl. 18 (4) (2020) 664

It can be shown that

Vσ = {x = (xj) : lim
i
tij(x) = ` uniformly in j,` = σ − lim x}

where,

tij(x) =
xσ(j) + xσ2(j) + · · · + xσi(j)

i + 1

Several authors including Raimi [19], Schaefer [20], Mursaleen and Edely [10], Mursaleen [12], Savaş [22, 23],

Nuray and Savaş [14], Pancaroǧlu and Nuray [16, 17] and some authors have studied invariant convergent

sequences. The concept of strongly σ-convergence was defined by Mursaleen [11]. Savaş and Nuray [24]

introduced the concepts of σ-statistical convergence and lacunary σ-statistical convergence and gave some

inclusion relations.

Now, we recall the basic concepts and some definitions and notations (See [1, 3–5, 7–9, 13, 15, 21]).

Let X be a normed space. For any given series
∑
i xi in X, let us consider the sets

S(
∑
i

xi) = {(ai) ∈ `∞ :
∑
i

aixi convergent}

Sw(
∑
i

xi) = {(ai) ∈ `∞ :
∑
i

aixi convergent for the weak topology}.

The above sets endowed with the sup norm and they will be called the space of convergence and the space

of weak convergence associated to the series
∑
i xi.

Definition 1.1. A series
∑
i xi in a normed space X is said to be a weakly unconditionally Cauchy(wuc)

if for each ε > 0 and f ∈ X∗, an n0 ∈ N can be found such that for each finite subset F ⊂ N with

F ∩{1, . . . ,n0} 6= ∅ is
∑
i∈F |f(xi)| < ε.

As a consequence,
∑
i xi is a wuc series in X if and only if each functional f ∈ X

∗ satisfies that∑∞
i=1 |f(xi)| < ∞.

In [18] it is proved that a normed space X is complete if and only if for every weakly unconditionally

Cauchy (wuc) series
∑
i xi, the space S(

∑
i xi) is also complete.

Diestel [6] proved the following characterization that will be used throughout the paper.

Theorem 1.1. Let
∑
i xi be a series in a normed space X. Then, the series

∑
i xi is wuc if and only if

there exists H > 0 such that



Int. J. Anal. Appl. 18 (4) (2020) 665

H = sup{‖
n∑
i=1

aixi‖ : n ∈ N, |ai| ≤ 1, i ∈{1, . . . ,n}}

= sup{‖
n∑
i=1

εixi‖ : n ∈ N,εi ∈{−1, 1}, i ∈{1, . . . ,n}}

= sup{
n∑
i=1

|f(xi)| : f ∈ BX∗}

where BX∗ is denotes the closed unit ball in X
∗

2. Main Results

Proposition 2.1. Let X be a normed space and (xn) an invariant convergent sequence in X. Then (xn) ∈

`∞(X).

Proof. Let (xn) be a sequence in X such that σ − limn xn = x0 for some x0 ∈ X. We can fix ε > 0 and

i0 ∈ N satisfying that ∥∥∥∥ 1i + 1
i∑

k=0

xσk(j)

∥∥∥∥ ≤‖x0‖ + ε
for every i ≥ i0 and j ∈ N. Then we have that for every j ∈ N is

‖xj‖ = ‖xσ0(j)‖ =
∥∥∥∥i0 + 2i0 + 1

i0+1∑
k=0

xσk(j)

i0 + 2
−
i0+1∑
k=1

xσk(j)

i0 + 1

∥∥∥∥ ≤
(
i0 + 2

i0 + 1
+ 1

)
(‖x0‖ + ε)

where the last term is a fixed constant, what concludes the proof. �

Definition 2.1. A series
∑
i xi in X is said to be invariant convergent to x0 ∈ X if σ − limn sn = x0,

where sn =
∑n
i=1 xi is sequence of partial sums, and we will denote it by Vσ −

∑
i xi = x0. Therefore,

Vσ −
∑
i xi = x0 if and only if

lim
i→∞

( j∑
k=1

xk +
1

i + 1

i∑
k=1

[
(i−k + 1)xσk(j)

])
= x0

uniformly in j ∈ N.

Definition 2.2. x0 is said to be weak invariant limit of a sequence (xn) if each function f ∈ X∗ verifies

that σ − lim f(xn) = f(x0) and we will write wσ − lim xn = x0.

Let X be a normed space and
∑
i xi a series in X. We define following sets:

Sσ(
∑
i

xi) = {(ai) ∈ `∞ : Vσ −
∑
i

aixi exists}

Swσ(
∑
i

xi) = {(ai) ∈ `∞ : wVσ −
∑
i

aixi exists}.

These spaces are the vector subspaces of `∞ and we consider them endowed with the sup norm.



Int. J. Anal. Appl. 18 (4) (2020) 666

Theorem 2.1. Let X be a Banach space and
∑
i xi a series in X. Then

∑
i xi is wuc(weakly unconditionally

Cauchy) if and only if Sσ(
∑
i xi) is complete.

Proof. Let
∑
i xi be a wuc series. We will prove that Sσ(

∑
i xi) is closed in `∞. Let (a

n) be a sequence in

Sσ(
∑
i xi), a

n = (ani ) for each n ∈ N and let also be a0 ∈ `∞ such that limn‖a
n −a0‖ = 0. We will show

that a0 ∈ Sσ(
∑
i xi). Let H > 0 be such that

H ≥ sup{‖
n∑
i=1

aixi‖ : n ∈ N, |ai| ≤ 1, i ∈{1, . . . ,n}}.

For each natural n there exists yn ∈ X such that yn = Vσ −
∑
i a
n
i xi. We will see that (yn) is a Cauchy

sequence.

If ε > 0 is given, there exists an n0 such that if p,q ≥ n0 ,then ‖ap −aq‖ <
ε

3H
. If p,q ≥ n0 are fixed,

there exists i ∈ N verifying

∥∥∥∥yp −
( j∑
k=1

a
p
kxk +

1

i + 1

i∑
k=1

[
(i−k + 1)ap

σk(j)
xσk(j)

])∥∥∥∥ < ε3 (2.1)

∥∥∥∥yq −
( j∑
k=1

a
q
kxk +

1

i + 1

i∑
k=1

[
(i−k + 1)aq

σk(j)
xσk(j)

])∥∥∥∥ < ε3 (2.2)
for each j ∈ N. Then, if p,q ≥ n0 we have that

‖yp −yq‖≤ (2.1) + (2.2) +
∥∥∥∥ j∑
k=1

(a
p
k −a

q
k)xk +

i∑
k=1

[
i−k + 1
i + 1

(a
p
σk(j)

−aq
σk(j)

)xσk(j)

]∥∥∥∥, (2.3)
where (2.3) ≤

ε

3
. Therefore, since X is Banach space, there exists y0 ∈ X such that limn‖yn −y0‖ = 0. We

will check that σ
∑
i a

0
ixi = y0, that is,

lim
i→∞

( j∑
k=1

a0kxk +
1

i + 1

i∑
k=1

[
(i−k + 1)a0σk(j)xσk(j)

])
= y0,

uniformly in j ∈ N.

If ε > 0 is given, we can fix a natural n such that ‖an −a0‖ <
ε

3H
and

‖yn −y0‖ <
ε

3
. Now, we can also fix i0 such that for every i ≥ i0 is

∥∥∥∥yn −
( j∑
k=1

ankxk +
1

i + 1

i∑
k=1

[
(i−k + 1)anσk(j)xσk(j)

])∥∥∥∥ < ε3



Int. J. Anal. Appl. 18 (4) (2020) 667

for every j ∈ N. Then, if i ≥ i0 it is satisfied that∥∥∥∥y0 −
( j∑
k=1

a0kxk +
1

i + 1

i∑
k=1

[
(i−k + 1)a0σk(j)xσk(j)

])∥∥∥∥ ≤‖y0 −yn‖
+

∥∥∥∥yn −
( j∑
k=1

ankxk +
1

i + 1

i∑
k=1

[
(i−k + 1)anσk(j)xσk(j)

])∥∥∥∥
+

∥∥∥∥ j∑
k=1

(an −a0)xk +
1

i + 1

i∑
k=1

[
(i−k + 1)(anσk(j) −a

0
σk(j))xσk(j)

]∥∥∥∥ ≤ 2ε3
+ ‖an −a0‖

(σ(j)∑
k=1

(ank −a
0
k)

‖an −a0‖
xk +

i∑
k=1

[
(i−k + 1)
i + 1

(an
σk(j)

−a0
σk(j)

)

‖an −a0‖
xσk(j)

])
≤

2ε

3
+

ε

3H
H ≤ ε

for every j ∈ N. Thus (a0n) ∈ Sσ(
∑
i xi).

Conversely, if Sσ(
∑
i xi) is closed, since c00 ⊂ Sσ(

∑
i xi), we deduce that c0 ⊂ Sσ(

∑
i xi). Suppose that∑

i xi is not wuc series. Then there exists f ∈ X
∗ verifying

∑∞
i=1 |f(xi)| = +∞.

We can choose a natural n1 such that
∑n1
i=1 |f(xi)| > 2.2 and for i ∈ {1, . . . ,n1} we define ai =

1

2
if

f(xi) ≥ 0 or ai =
−1
2

if f(xi) < 0.

There exists n2 > n1 such that
∑n2
i=n1+1

|f(xi)| > 3.3 and for i ∈ {n1 + 1, . . . ,n2} we define ai =
1

3
if

f(xi) ≥ 0 or ai =
−1
3

if f(xi) < 0.

In this manner we obtain an increasing sequence (nk)k in N and a sequence a = (ai)i in c0 such that∑∞
i=1 aif(xi) = +∞. Since (ai)i ∈ Sσ(

∑
i xi), it follows that σ

∑
i aixi exists and therefore

(∑n
i=1 aif(xi)

)
n

is bounded sequence, which is a contradiction. �

Then we have the following result.

Corollary 2.1. Let X be a Banach space and
∑
i xi a series in X. Then

∑
i xi is a wuc(weakly uncondi-

tionally Cauchy) series if and only if for each sequence (ai)i ∈ c0 it is satisfied that Vσ −
∑
i aixi exists.

Proof. Let
∑
i xi be a wuc series in X. Then, we have that Sσ(

∑
i xi) is complete. Since c00 ⊂ Sσ(

∑
i xi),

we deduce that c0 ⊂ Sσ(
∑
i xi), that is, Vσ −

∑
i aixi exists for every sequence (ai) ∈ c0. The converse is

proved similar to the end of the previous demonstration. �

Remark 2.1. Let X be a normed space and
∑
i xi a series in X. We consider the linear map T :

Sσ(
∑
i xi) → X defined by T(a) = Vσ −

∑
i aixi.



Int. J. Anal. Appl. 18 (4) (2020) 668

Suppose that
∑
i xi is a wuc series and consider H = sup{‖

∑n
i=1 aixi‖ : n ∈ N, |ai| ≤ 1, i ∈ {1, . . . ,n}}.

Then, it is easy to check that if a ∈ Sσ(
∑
i xi) then ‖T(a)‖ = ‖Vσ −

∑
i aixi‖ ≤ H‖a‖ and therefore T is

continuous.

Conversely if T is continuous and {a1, . . . ,aj} ⊂ [−1, 1], it is satisfied that ‖
∑j
i=1 aixi‖ = ‖Vσ −∑∞

i=1 aixi‖≤‖T‖ (considering ai = 0 if i > j), which implies that
∑
i xi is a wuc series.

In the next theorem we study the completeness of space Swσ(
∑
i xi).

Theorem 2.2. Let X be a Banach space and
∑
i xi a series in X. Then

∑
i xi is a wuc series if and only

if Swσ(
∑
i xi) is complete.

Proof. Consider
∑
i xi to be a wuc series. It will be enough to prove that Swσ(

∑
i xi) is closed in `∞. Let (a

n)

be sequence in Swσ(
∑
i xi), a

n = (ani )i for each n ∈ N and let also be a
0 ∈ `∞ such that limn‖an −a0‖ = 0.

We will show that a0 ∈ Swσ(
∑
i xi). Let H > 0 be such that

H ≥ sup{‖
n∑
i=1

aixi‖ : n ∈ N, |ai| ≤ 1, i ∈{1, . . . ,n}}

For each natural n there exists yn ∈ X such that yn = wVσ−
∑
i a
n
i xi. We will check that (yn)n is Cauchy

sequence.

If ε > 0 is given, there exists an n0 such that if p,q ≥ n0 ,then ‖ap − aq‖ <
ε

3H
. We fix p,q ≥ n0

and we have that there exists f ∈ SX∗ (unit sphere in X∗)verifying ‖yp − yq‖ = |f(yp − yq)|. Since

Vσ −
∑
i a
p
i f(xi) = f(yp) and Vσ −

∑
i a
q
if(xi) = f(yq), there exists i ∈ N such that∣∣∣∣f(yp) −

( j∑
k=1

a
p
kf(xk) +

1

i + 1

i∑
k=1

[
(i−k + 1)ap

σk(j)
f(xσk(j))

])∣∣∣∣ < ε3 (2.4)
∣∣∣∣f(yq) −

( j∑
k=1

a
q
kf(xk) +

1

i + 1

n∑
k=1

[
(i−k + 1)aq

σk(j)
f(xσk(j))

])∣∣∣∣ < ε3 (2.5)
for each j ∈ N. Then, if p,q ≥ n0 we have that

‖yp −yq‖ =|f(yp) −f(yq)| ≤ (2.4) + (2.5) (2.6)

+

∣∣∣∣ j∑
k=1

(a
p
k −a

q
k)f(xk) +

i∑
k=1

[
i−k + 1
i + 1

(a
p
σk(j)

−aq
σk(j)

)f(xσk(j))

]∣∣∣∣, (2.7)
where (2.6) ≤

ε

3
. Therefore, since X is Banach space, there exists y0 ∈ X such that limn‖yn −y0‖ = 0.

We will check that wVσ −
∑
i a

0
ixi = y0.

If ε > 0 is given, we can fix a natural n such that ‖an −a0‖ <
ε

3H
and

‖yn −y0‖ <
ε

3
. Consider a functional f ∈ BX∗ . We have that there exists i0 ∈ N such that if i ≥ i0 is



Int. J. Anal. Appl. 18 (4) (2020) 669

∣∣∣∣f(yn) −
( j∑
k=1

ankf(xk) +
1

i + 1

i∑
k=1

[
(i−k + 1)anσk(j)f(xσk(j))

])∣∣∣∣ < ε3
for every j ∈ N. Then, if i ≥ i0 and j ∈ N, we have that∣∣∣∣f(y0) −

( j∑
k=1

a0kf(xk) +
1

i + 1

i∑
k=1

[
(i−k + 1)a0σk(j)f(xσk(j))

])∣∣∣∣ ≤ |f(y0 −yn)|
+

∣∣∣∣f(yn) −
( j∑
k=1

ankf(xk) +
1

i + 1

i∑
k=1

[
(i−k + 1)anσk(j)f(xσk(j))

])∣∣∣∣
+

∣∣∣∣ j∑
k=1

(an −a0)f(xk) +
1

i + 1

i∑
k=1

[
(i−k + 1)(anσk(j) −a

0
σk(j))f(xσk(j))

]∣∣∣∣ ≤ ε
that is, wVσ −

∑
i a

0
ixi = y0 and a

0 ∈ Swσ(
∑
i xi).

Conversely, if Swσ(
∑
i xi) is complete, which implies that c0 ⊂ Swσ(

∑
i xi). Suppose that there exists

f ∈ X∗ verifying
∑∞
i=1 |f(xi)| = +∞.

Then, as we did in Theorem 2.1, a sequence a = (ai) in c0 can be obtained such that
∑
i aif(xi) = +∞

since a ∈ Swσ(
∑
i xi), there will exists x0 ∈ X such that wVσ−

∑
i aixi = x0 and it will be Vσ−

∑
i aif(xi) =

x0. But this implies that the sequence

(∑n
i=1 aif(xi)

)
n

is bounded which is a contradiction. �

Remark 2.2. Let X be a Banach space
∑
i xi a series in X. We consider the linear map T:Swσ(

∑
i xi) → X

defined by T(a) = wVσ −
∑
i aixi. We will show that

∑
i xi is wuc series if and only if T is continuous.

We define H = sup{‖
∑n
i=1 aixi‖ : n ∈ N, |ai| ≤ 1, i ∈ {1, . . . ,n}} and take a ∈ Swσ(

∑
i xi). Then

wVσ−
∑
i aixi = x0 exists and we can take f ∈ SX∗ such that |T(a)| = |f(T(a))| = |Vσ−

∑
i aif(xi)| ≤ H‖a‖.

Conversely if T is continuous.Then if {a1, . . . ,an}⊂ [−1, 1], we have that

‖
∑n
i=1 aixi‖ = ‖wVσ −

∑∞
i=1 aixi‖ ≤ ‖T‖ (considering ai = 0 if i > n), which implies that

∑
i xi is a wuc

series.

From the previous theorem and its proof the following corallary can be easily proved.

Corollary 2.2. Let X be a Banach space
∑
i xi a series in X. Then the following are equivalent:

(1)
∑
i xi is a wuc series.

(2) Swσ(
∑
i xi) is complete.

(3) c0 ⊂ Swσ(
∑
i xi) (wVσ −

∑
i aixi exists for each a = (ai) ∈ c0).

Let us see that the hypothesis of completeness in the two previous theorems is completely necessary.

Let X be a non-complete normed space. Then it is easy to prove that there exists a sequence
∑
i xi in

X such that ‖xi‖ <
1

i2i
and

∑
i xi = x

∗∗ ∈ X∗∗\X. Then we have that Vσ −
∑
i xi = x

∗∗. If we consider



Int. J. Anal. Appl. 18 (4) (2020) 670

the series
∑
i zi defined by zi = ixi for each n ∈ N, we have that

∑
i zi is wuc series. Consider the sequence

a = (ai) ∈ c0 given by ai =
1

i
. It is satisfied that Vσ −

∑
i aizi ∈ X

∗∗\X and therefore a /∈ Sσ(
∑
i zi) and

a /∈ Swσ(
∑
i zi).

Let X be a normed space and X∗ its dual space. Let also
∑
i fi be a series in X

∗. It is known that [6],∑
i fi is wuc if and only if

∑
i |fi(x)| < ∞ for each x ∈ X.

Now we consider the vector space

S∗wσ(
∑
k

fi) = {a = (ai) ∈ `∞ : ∗wVσ −
∑
i

aifi exists}

, where ∗wVσ −
∑
i aifi = f0 if and only if Vσ −

∑
i aifi(x) = f0(x) for each x ∈ X.

Theorem 2.3. Let X be a normed space. It is satisfied that 1 ⇒ 2 ⇒ 3, where:

(1)
∑
i f(i) is a wuc series.

(2) S∗wσ(
∑
i fi) = `∞.

(3) If x ∈ X and M ⊂ N, then Vσ −
∑
i∈M fi(x) exists.

Besides, if X is a barelled normed space, the three items are equivalent.

Proof. From the ∗ weak compacity of BX∗ we deduce that 1 ⇒ 2. It is clear that 2 ⇒ 3.

We suppose now that X is barelled and we will prove that 3 ⇒ 1. Effectively, our goal is to prove

that E = {
∑n
i=1 aifi : n ∈ N, |ai| ≤ 1, i ∈ {1, . . . ,n}} is pointwise bounded for each x ∈ X and therefore

E is bounded, which implies that
∑
i fi is wuc series. Suppose that E is not pointwise bounded, that

is, there exists x0 ∈ X such that
∑
i |fi(x0)| = +∞. Then, we can choose a subset M ⊂ N such that∑

i∈M fi(x0) = + −∞. But, by hypothesis, Vσ −
∑
i∈M fi(x0) exists, which is a contradiction. �

When σ(j) = j + 1, we have the almost all definitions and theorems in [2] concerning almost summability.

Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication

of this paper.

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	1. Introduction
	2. Main Results
	References