Orlicz–Pettis Theorem through Summability Methods

Abstract

This paper unifies several versions of the Orlicz–Pettis theorem that incorporate summability methods. We show that a series is unconditionally convergent if and only if the series is weakly subseries convergent with respect to a regular linear summability method. This includes results using matrix summability, statistical convergence with respect to an ideal, and other variations of summability methods.

Schur Lemma and Uniform Convergence of Series through Convergence Methods

“…The notion of induced summability was introduced in [10] and it allows us to unify several results that incorporate different types of weak convergence. Let ρ : D ρ ⊂ R N → R be a summability method on R. The summability method ρ could induce a summability method R on every normed space X as follows.…”

“…In this way, it is possible to see a classical result from a new point of view. Sometimes [10,11], it is possible to characterize those summability methods for which these classical results hold. For instance, in [10], the summability methods for which the classical Orlicz-Pettis's result is true are characterized, namely, it is possible to obtain a version of the Orlicz-Pettis's theorem for any regular convergence method.…”

“…We continued the research line started in [10,11], and we aim to unify different versions of Swartz's result [5,7,15]. For instance, Schur type results were obtained for any regular matrix summability method [15] and for the Banach-Lorentz convergence [5].…”

Riesz multiplier convergent spaces of operator valued series and a version of Orlicz- Pettis theorem

“…Kama and Altay [10] obtained some new multiplier space by using Fibonacci sequence spaces, and Kama et al [11] gave similar results for multiplier spaces which were obtained from backward difference matrix. Most recently, León-Saavedra et al [17,18] also obtained some new versions of Orlicz-Pettis theorem by using w p -summability and a general summability methods.…”

“…The Orlicz-Pettis Theorem is one of the important results in the Theory of Functional Analysis. Many generalizations and applications of this theorem can be found in [5,17,18,22,24,25,27]. Before stating and proving the Orlicz-Pettis Theorem by means of Riesz convergence, we will give the following definition (see [22]):…”