λ-Rearrangements Characterization of Pringsheim Limit Points

In [1] Agnew presented the following theorem: if x is a bounded sequence and A is a regular summability matrix, then there exists a subsequence y of x such that each limit point of x is a limit point of Ay. Fridy [2] extended this result by replacing subsequence with rearrangement. Keagy [3] presented two theorems that strengthened the results of both Agnew and Fridy. This was accomplished by weakening the regularity conditions and replacing finite limit point for bounded sequence. The goal of the paper is to present two multidimensional theorems analogou to Keagy’s theorems with λ-rearrangement replacing rearrangement, RH-regularity replacing regularity, and convergent in the Pringsheim sense replacing convergent. Other implications will also be presented.


Introduction
In [1] Agnew presented the following theorem: if x is a bounded sequence and A is a regular summability matrix, then there exists a subsequence y of x such that each limit point of x is a limit point of Ay.Fridy [2] extended this result by replacing subsequence with rearrangement.Keagy [3] presented two theorems that strengthened the results of both Agnew and Fridy.This was accomplished by weakening the regularity conditions and replacing finite limit point for bounded sequence.The goal of the paper is to present two multidimensional theorems analogou to Keagy's theorems with λ-rearrangement replacing rearrangement, RH-regularity replacing regularity, and convergent in the Pringsheim sense replacing convergent.Other implications will also be presented.

Definitions, notations, and preliminary results
Definition 2.1.Let A denote a four-dimensional summability method that maps the complex double sequences x into the double sequence Ax, where the mnth term to Ax is as follows: (Ax) m,n = ∞,∞ k,l=1,1 a m,n,k,l x k,l .
(2.1) Definition 2.2 (see Pringsheim [4]).A double sequence x = [x k,l ] has a Pringsheim limit L (denoted by P-limx = L) provided that given > 0 there exists N ∈ N such that |x k,l − L| < whenever k,l > N.Such an x will be more briefly described as "P-convergent." In addition to P-convergent, Pringsheim also presented the following notion of divergent.
Definition 2.3.A double sequence x is called definite divergent if for every (arbitrarily large) G > 0 there exist two natural numbers n 1 and In [5], Robison presented a four-dimensional notion of regularity for double sequences with an additional assumption of boundedness.This assumption was made because a double sequence which is P-convergent is not necessarily bounded.In addition to this notion, Robison and Hamilton both presented a Silverman-Toeplitz-type multidimensional characterization of regularity in [5,6].The definition of the regularity for fourdimensional matrices will be stated next, followed by the Robison-Hamilton characterization of the regularity of four-dimensional matrices.
Definition 2.4.The four-dimensional matrix A is said to be RH-regular if it maps every bounded P-convergent sequence into a P-convergent sequence with the same P-limit.The following definition of the subsequence of a double sequence was presented in [7].
Definition 2.6.The double sequence [y] is a double subsequence of the sequence [x] provided that there exist two increasing double-index sequences {n j } and {k j } such that if z j = x nj ,kj , then y is formed by Using this concept of subsequence, the following definitions for Pringsheim limit points and divergence of double sequences were presented in [7].Definition 2.7.A number β is called a Pringsheim limit point of the double sequence x = [x n,k ] provided that there exists a subsequence y = [y n,k ] of [x n,k ] that has Pringsheim limit β : P-lim y n,k = β.Richard F. Patterson 3 Definition 2.8.A double sequence x is divergent in the Pringsheim sense (P-divergent) provided that x does not converge in the Pringsheim sense (P-convergent).
In addition to these definitions, the author also presented the following theorem in [8].
Theorem 2.9.If each of T and A is an RH-regular matrix, x is any bounded double-complex sequence, and is any bounded positive term double sequence with P-lim k,l k,l = 0, then there exists a subsequence y of x such that T(Ay) exists and each P-limit of x is a P-limit of T(Ay).
In [9], Patterson and Rhoades presented the following definition for rearrangement of double sequences.Definition 2.10.Fix λ > 1.The double sequence y (π,λ) is called a "λ-rearrangement" of the double sequence x provided that there is a one-to-one function π from the positive integers into themselves such that (1) {z i (m,n) } is a one-dimensional sequence constructed from the double sequence {x m,n } as follows: (2.4) The word λ-permutation will be reserved to indicate the reordering of a finite double sequence.In addition, we will say that the ordered pair (k,l) is in the λ-wedge [10]

Main results
The theorems presented in this section are multidimensional analogs of Keagy theorems in [3].Throughout the proofs of the main results, we will use the ordering presented in Definition 2.6.Theorem 3.1.Let A be a four-dimensional matrix transformation with P-null pairwise rows and pairwise columns.Also let x be a bounded double-complex sequence.If y is a double Pringsheim subsequence of x such that Ay exists and has a finite P-limit point, then there exists a λ-rearrangement z of x such that Az exists and each P-limit point of Ay is a P-limit point of Az.
Proof.In [8], Theorem 2.9 grants us a double sequence v constructed from u such that each term of u is a P-limit point of T(Ay) and each P-limit point is a term of v.In addition, v has the property that each term of u is not only a P-limit point and/or a term of T(Ay) but also a term in the λ-wedge of v. Let us choose the order pairs (m 1 ,n 1 ) and (α 1 ,β 1 ).Also let us denote the λ-permutation of {x k,l }'s with 1 ≤ k ≤ α 1 and 1 ≤ l ≤ β 1 as B α1,β1 (i.e., α 1 β 1 -block).In addition, let (k,l) be the first (β 2 + α 2 − β 1 − α 1 − 1)-terms of {x i, j } with the following properties: (1) (k,l) is in the λ-wedge where the first (β 2 + α 2 − β 1 − α 1 + 1)-terms has an ordering as in Definition 2.6.Also let us choose (m 2 ,n 2 ) and (α 2 ,β 2 ) such that m 2 > m 1 , n 2 > n 1 , α 2 > α 1 , and β 2 > β 1 , where (α 1 ,β 2 ) and (α 2 ,β 1 ) are in the λ-wedge of v such that a m,n,α1, j z α1, j − y α1, j < 1 2 4 , sup Let us define z as follows: and z k,l = x k,l otherwise.Suppose in general that the doubleindex sequences (α s−1 ,β t−1 ) and (m s−1 ,n t−1 ) have been chosen with In addition, let (k,l) be the first (β t + α s − β t − α s − 1)-terms of {x i, j } with the following properties: (1) (k,l) is in the λ-wedge of {x i, j }, (2) (k,l) is not in the α s β t -block of {x i, j }, (3) (k,l) is not in {y i, j : α s < i < ∞ ∪ β t < j < ∞}, where the first (β t + α s − β t−1 − α s−1 − 1)-terms have an ordering as in Definition 2.6.Now let us choose (m s ,n t ) and (α s ,β t ) such that m s > m s−1 , n t > n t−1 , α s > α s−1 , and β t > β t−1 , where (α s−1 ,β t ) and (α s ,β t−1 ) are in the λ-wedge of v with the following properties: Let us define z as follows: and z k,l = x k,l otherwise.Let us consider the following: (3.4)Therefore, each P-limit point of Ay is a P-limit of Az.This completes the proof of this theorem.
Theorem 3.2.If x is a double-complex sequence and A is a row pairwise-finite four-dimensional matrix satisfying conditions RH 1 through RH 5 of RH-regularity, then there exists a λ-rearrangement y of x such that every limit point of x (finite or infinite) is a limit point of (Ay).

Corollary 3 . 3 .
{z ζ,η }, where {z ζ,η } are selected such that {ζ, η} corresponds to the first index of x in x\{z i, j } αr −1, βs−1 i, j=1,1 such that αr ,βs k,l=1,1 a mr ,ns,k,l z k,l > 2 r+s .(3.14)This process grants us two positive integer double sequences and a λ-rearrangement z of x having the following properties: |(Az) mr ,ns − v r,s | = o(1) and {|(Az) mr ,ns |} which is definite divergent.This completes the proof.If there exists a four-dimensional matrix A satisfying RH 1 through RH 5 such that Az is definite divergent for every λ-rearrangement z of x, then x is definite divergent.Corollary 3.4.If there exists a four-dimensional matrix A satisfying RH 1 through RH 5 such that Az is bounded for every λ-rearrangement z of x, then x has only bounded subsequence.