1. Introduction
Let X be a normed space. A linear summability method on X is a rule to assign limits to a sequence, that is, it is a linear map . A summability method is said to be regular if, for each convergent sequence in X, that is, , we have that .
The methods of summability were born at the beginning of the 20th century, with the development of the theory of Fourier Analysis. For example, statistical convergence and strong Cesàro convergence were introduced respectively by Zygmund [
1] and Hardy [
2], and both concepts were surprisingly connected thanks to the work of Connor [
3] fifty years later. Since then, the Summability Theory has taken on a life of its own, with deep and beautiful results (see the recent monographs [
4,
5] for historical notes). Moreover, the theory has important applications on Applied Mathematics (see the recent monograph by Mursaleen [
6]).
The Orlicz–Pettis Theorem is a classic result concerning a convergent series, so beautiful that it has attracted the interest of many mathematicians and it has been strengthened and generalized in many directions. An early survey is Kalton’s paper [
7]. The reader can see in [
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19] recent results about the Orlicz–Pettis type Theorems.
Let us recall that a series in a Banach space X is said to be unconditionally convergent (u.c) if, for each permutation of the natural numbers , we have that is convergent. A series is weak-subseries convergent if, for any , there exists such that the partial sums converges weakly to (here denotes the characteristic function on M). The classical Orlicz–Pettis Theorem states that a series is unconditionally convergent if and only if is weakly subseries convergent.
Diestel and Faires sharpened the classical result by Orlicz–Pettis in the following sense ([
20] I.4.7). Let
X be a Banach space that contains no copy of
and let
be a total subset of
. Then, a formal series
in
X such that every subseries is
-convergent, that is, for each subset
, there exists
, such that
for all
; then,
is norm unconditionally convergent.
On the other hand, attempts have been made to replace weak convergence with other (weak) summability method. At this point, we find several results in the literature. For instance, the Orlicz–Pettis Theorem remains true if we replace the weak convergence by the weak statistical convergence (see [
8]). It is also true for the weak-Cesàro convergence [
21], for the weak
A-convergence,
A being an infinite matrix with non-negative entries [
22], for the weak-statistical Cesàro convergence [
9], and more recently for the
-strong Cesàro convergence [
15]. In this note, we aim to unify all known results, obtaining an Orlicz–Pettis Theorem for a general summability method; of course, we need to place some limits on the summability method because the result fails for an arbitrarily summability method. Namely, the result is true for any linear regular summability method. It is surprising how we can weaken the weak convergence hypothesis in the Orlicz–Pettis result by almost any other weaker summability method. The paper is organized as follows: in
Section 2, we will show a General Orlicz–Pettis Theorem for summability methods. Next, we will see how effectively our result unifies the known results, and we will see new applications.
2. Main Results
Let be a linear summability method, that is, a subset and a linear function : , which assigns a unique real number to a sequence . In addition, is said to be regular, if, for every convergent sequence , the sequence , -converge to the same limit. A summability method induces a weak summability method in X as follows: a sequence is -convergent to if and only if is -convergent to for all . Let us observe that, in general, the convergence method could be degenerate, that is, . However, if is regular, then is non-empty; moreover, is also regular.
Proposition 1. If ρ is regular, then is regular.
Proof. Let us suppose that ; then, for each , , since is regular . Therefore, as desired. □
Theorem 1. Let X be a real Banach space, ρ a regular summability method on and the summability method induced by ρ. Then, a series is unconditionally convergent if and only is -subseries convergent in X.
Proof. Let be an unconditionally convergent series, and let . By applying the classical Orlicz–Pettis Theorem, we obtain that there exists such that the sequence weakly converges to , that is, for each , we have that the sequence . Since is regular, we have that for each , that is, -converges to , as desired.
Now, let us suppose that, for any
, there exists
such that
-converges to
. First of all, we will prove that
is a weakly unconditionally Cauchy series. If not, let us argue by contradiction, so let us suppose that there exists
such that
. Let us consider the following subsets
and
. In addition, let us define the sequence
then
, hence the sequence
does not
converge to any
. On the other hand, by hypothesis, given
, there exists that
such that
and
. Therefore,
a contradiction. Therefore, for any
, we have
.
Now, let us show that, given , there exists such that weakly converges to . Let , since , we deduce that the series is convergent to some , and hence -convergent to . On the other hand, by hypothesis, there exists such that , that is, for each , we have that . Therefore, . Hence, we obtain that, for any , the sequence converges to , that is, weakly converges to . Thus, by applying the classical Orlicz–Pettis Theorem, we obtain that the series is unconditionally convergent as desired. □
Remark 1. The following example was pointed out by one of the referees. Let us consider the following linear summability method: ρ, a sequence is said to be ρ-convergent to if . Then, it is clearly in the realm of bounded sequences . Thus, ρ is not regular. Now, let us consider on the summability method induced by ρ, which we denote by . For every , we have that . However, is not norm convergent to 0. The argument of the proof breaks down if we can’t guarantee that . This fact highlights the importance of regularity in the proof of the above result.
Now, let us see some applications of Theorem 1. We will say that is a non-trivial ideal if
and .
If , then .
If and , then .
Additionally, we say that is regular (or admissible) if it contains all finite subsets.
A non-trivial regular ideal
defines a regular summability method on any metric space. We will say that a sequence
is
-convergent to
(in short
) if, for any
, the subset
Thus, given a Banach space X, the -convergence defines a weakly summability method in X; we will say that a sequence is weakly- convergent to if and only if, for any , we have .
Corollary 1. Let be a non-trivial ideal. Then, a series is a real Banach space; X is unconditionally convergent if and only if is subseries weakly- convergent.
In particular, if we consider the ideal
of all subsets in
with zero density, the ideal convergence induced by
(which is non trivial and regular) is the statistical convergence. Therefore, the above Corollary is also true for the weak-statistical convergence [
8].
Now, let us consider a regular matrix summability method induced by an infinite matrix , which is defined as follows. A sequence is A-summable to L if . A matrix A is regular if the usual convergence implies the A-convergence, and the limits are preserved. Now, if X is a Banach space, then the matrix A also induces a summability method on a Banach space X; we say that a sequence is A-convergent to if . The matrix A also induces a weakly convergence, a sequence is weakly A-convergent to if, for any , we have that is A-convergent to . Applying Theorem 1, we get:
Corollary 2. Let A be a regular matrix. Then, a series is unconditionally convergent if and only if is subseries weak-A-convergent.
Thus, we obtain the results in [
22]. In particular, if
A is the Cesàro matrix, we obtain the results in [
21].
Let us consider the following summability method : we will say that a sequence is Cesàro, statistically convergent to L, if the Cesàro means is statistically convergent to L. Given a Banach space X, the summability method induces a weakly summability method in X. Namely, we say that a sequence is weakly statistically Cesàro convergent to if, for every , the sequence is Cesàro statistically convergent to .
As a consequence, we obtain the results in [
9].
Corollary 3. Let A be a regular matrix. Then, a series is unconditionally convergent if and only if is subseries weakly statistically Cesàro-convergent.
In brief, Theorem 1 not only unifies the known results, but also can be widely applied to several summability methods obtaining new versions of the Orlicz–Pettis theorem. For instance, it applies for the Erdös–Ulam convergence,
-Cesàro convergence,
f-statistical convergence, etc. (see [
15,
23]).