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CUBO A Mathematical Journal
Vol.17, No¯ 02, (89–95). June 2015

Submatrices of Four Dimensional Summability Matrices

Fatih Nuray

Department of Mathematics,

AfyonKocatepe University, Afyonkarahisar, Turkey,

Deparment of Mathematics and Statistics, University of North Florida, Jacksonville, FL, USA

University of North Florida, Jacksonville, FL, USA

fnuray@aku.edu.tr

Richard F. Patterson

Department of Mathematics and Statistics,

University of North Florida Jacksonville,

Florida, 32224

rpatters@unf.edu

ABSTRACT

In this paper, we show that a matrix that maps ℓ′′ into ℓ′′ can be obtained from any

RH-regular matrix by the deletion of rows. Also a four dimensional conservative matrix

can be obtained by the deletion of rows from a matrix that preserves boundedness. We

will use these techniques to derive a sufficient condition for a four dimensional matrix

to sum an unbounded sequence.
RESUMEN

En este trabajo probamos que una matriz que lleva ℓ′′ en ℓ′′ se puede obtener a partir de

cualquier matriz RH-regular eliminando filas. También una matriz cuatro dimensional

conservative se puede obteber eliminando filas en una matriz que preserva acotación.

Usamos estas técnicas para encontrar una condición suficiente para que una matriz

cuatro dimensional sume una sucesión no acotada.

Keywords and Phrases: Submatrix, four dimensional summability matrix, Double Sequences,

Pringsheim Limit, RH-regular matrix

2010 AMS Mathematics Subject Classification: 40C05, 40D05.



90 Fatih Nuray & Richard F. Patterson CUBO
17, 2 (2015)

1 Introduction

The most well-known notion of convergence for double sequences is the convergence in the sense

of Pringsheim. Recall that a double sequence x = {xk,l} of complex (or real) numbers is called

convergent to a scalar ℓ in Pringsheim’s sense (denoted by P-lim x = ℓ) if for every ǫ > 0 there

exists an N ∈ N such that |xk,l − ℓ| < ǫ whenever k, l > N. Such an x is described more briefly

as ”P-convergent”. It is easy to verify that x = {xk,l} convergences in Pringsheim’s sense if and

only if for every ǫ > 0 there exists an integer N = N(ǫ) such that |xi,j − xk,l| < ǫ whenever

min{i, j, k, l} ≥ N. A double sequence x = {xk,l} is bounded if there exists a positive number M

such that |xk,l| ≤ M for all k and l, that is, if supk,l |xk,l| < ∞.

A double sequence x = {xk,l} is said to convergence regularly if it converges in Pringsheim’s sense

and, in addition, the following finite limits exist:

lim
k→∞

xk,l = ℓl (l = 1, 2, ...),

lim
l→∞

xk,l = Lk (k = 1, 2, ...).

Note that the main drawback of the Pringsheim’s convergence is that a convergent sequence fails

in general to be bounded. The notion of regular convergence lacks this disadvantage.

A double sequence x is divergent in the Pringsheim sense(P-divergent) provided that x does not

convergence in the Pringsheim sense.

Let A = (am,n,k,l) denote a four dimensional summability method that maps the complex double

sequence x into the double sequence Ax where the mn-th term to Ax is as follows:

(Ax)m,n =

∞∑

k=0

∞∑

l=0

am,n,k,lxk,l.

In [12] Robison presented the following notion of regularity for four-dimensional matrix transfor-

mation and a Silverman-Toeplitz type characterization of such notion.

Definition 1.1. 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 assumption of bounded was added because a double sequence which is P-convergent is not

necessarily bounded.

The four-dimensional matrix A is said to be RH-conservative if it maps every bounded P-

convergent sequence into a P-convergent sequence. In [12], Robison presented the following notion

of a conservative four-dimensional matrix transformation and a SilvermanToeplitz type character-

ization of such a notion.

Theorem 1.2. ([5],[12]) The four-dimensional matrix A is RH-conservative if and only if

RH − C1: P-limm,n am,n,k,l = ckl for each k and l;



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Submatrices of Four Dimensional Summability Matrices 91

RH − C2: P-limm,n
∑

∞,∞
k,l=0,0

am,n,k,l = α;

RH − C3: P-limm,n
∑

∞

k=0
|am,n,k,l| = 0 for each l;

RH − C4: P-limm,n
∑

∞

l=0
|am,n,k,l| = 0 for each k;

RH − C5:
∑

∞,∞
k,l=0,0

|am,n,k,l| is P-convergent;

RH − C6: there exist finite positive integers ∆ and Γ such that
∑

k,l>Γ
|am,n,k,l| < ∆.

Along these same lines, Robison and Hamilton presented a Silverman-Toeplitz type multidimen-

sional characterization of regularity in [5] and [12].

Theorem 1.3. (Hamilton [5], Robison [12]) The four-dimensional matrix A is RH-regular if and

only if

RH1: P-limm,n am,n,k,l = 0 for each k and l;

RH2: P-limm,n
∑

∞,∞
k,l=0,0

am,n,k,l = 1;

RH3: P-limm,n
∑

∞

k=0
|am,n,k,l| = 0 for each l;

RH4: P-limm,n
∑

∞

l=0
|am,n,k,l| = 0 for each k;

RH5:
∑

∞,∞
k,l=0,0

|am,n,k,l| is P-convergent;

RH6: there exist finite positive integers ∆ and Γ such that
∑

k,l>Γ
|am,n,k,l| < ∆.

The set of all absolutely convergent double sequences will be denoted ℓ′′, that is

ℓ′′ = {x = {xk,l} :

∞∑

m=0

∞∑

n=0

|xk,l| < ∞}.

In [9], Patterson proved that the matrix A = (am,n,k,l) determines an ℓ
′′ − ℓ′′ method if and

only if

sup
k,l

∞∑

m=0

∞∑

n=0

|am,n,k,l| < ∞. (1)

In this paper, firstly we will obtain a general correspondance between RH-regular matrices and

ℓ′′ − ℓ′′ matrices by proving that every RH-regular matrix gives rise to an ℓ′′ − ℓ′′ matrix by the

removing of appropriate rows. Secondly we prove that a matrix from double bounded sequences

to double bounded sequences contains a row-submatrix is conservative. Lastly, in order to prove

a criterion for the summability of an unbounded double sequence the row-selection technique will

be replaced by a column-selection technique.

2 Main Results

Theorem 2.1. If A = (amnkl) is a four dimensional summability matrix in which each row and

each column converge to zero and

sup
m,n

|amnkl| = α < ∞



92 Fatih Nuray & Richard F. Patterson CUBO
17, 2 (2015)

for each k and l then A contains a row-submatrix that is an ℓ′′ − ℓ′′ matrix.

Proof. First chose positive integer v0 and w0 satisfying |av0,w0,0,0| ≤ 1; then, using the assumption

that

P − lim
k,l

av0,w0,0,0 = 0,

choose k0 and l0 so that k > k0 and l > l0implies |av0,w0,k,l| ≤ 1. Having selected vi, wj, ki and

lj for i < p and j < q we choose vp > vp−1 and wq > wq−1 so that

k ≤ kp−1, l ≤ lq−1 implies |avp,wq,k,l| ≤ 2
−(p+q)

then choose kp > kp−1 and lq > lq−1 so that

k > kp, l > lq implies |avp,wq,k,l| ≤ 2
−(p+q).

Now define the submatrix B by bmnkl = avp,wq,k,l. The above construction guarantees that each

column sequence of B is dominated, except for at most one term, by the sequence (2−(p+q)); that

is, if kp−1 < k ≤ kp, lq−1 < l ≤ lq, i 6= p and j 6= q, then |bijkl| = |avi,wj,k,l| ≤ 2
−(i+j). Since

|avp,wq,k,l| ≤ α, it is clear that for each k, l,

∞∑

p=0

∞∑

q=0

|bpqkl| ≤ 4 + α.

Hence, by (1.1), B is an ℓ′′ − ℓ′′ matrix.

Corollary 2.2. Every four dimensional RH-regular matrix contains a row-submatrix that is an

ℓ′′ − ℓ′′ matrix.

Theorem 2.3. If A maps double bounded sequences into itself, then A contains a conservative

row-submatrix B.

Proof. Since A maps double bounded sequences into itself, we have

sup
m,n

∑

k

∑

l

|amnkl| < ∞.

Therefore the sequence of row sums (
∑

k

∑
l
amnkl) is bounded, so we can choose a convergent

subsequence. This yields a row-submatrix A′ of A that satisfies properties (ii) and (iii). It remains

to choose a row-submatrix of A′ whose columns are convergent sequences. But this is a special

case of the familiar diagonal process that is used in the proof of the multidimensional analog of

Helly Selection Principle (see [13],[2])for we have a family of functions (the rows of A′) that are



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Submatrices of Four Dimensional Summability Matrices 93

uniformly bounded by supm,n
∑

k

∑
l
|amnkl| on their countable domain






(0, 0), (0, 1), (0, 2), (0, 3), (0, 4), ...

(1, 0), (1, 1), (1, 2, ) (1, 3), (1, 4), ...

(2, 0, ) (2, 1), (2, 2), (2, 3), (2, 4), ...
...

...
...

...
...

...






Therefore we can select a sequence of these functions that converges at each k and l. This

sequence of rows of A′ are then the rows of B.

3 Summability of Unbounded Sequences

In [8] Patterson and Savaş proved multidimensional generalization of Agnew’s theorem.

Theorem 3.1. If the four dimensional matrix A = (amnkl) such that

∑

k

∑

l

|amnkl| < ∞ and P − lim
m,n→∞

sup
k,l

|amnkl| = 0, (2)

then there exists at least one P-divergent double sequence of zeros and ones that is A summable .

By modifying the proof of Theorem 2.1 from row selection to column selection, we can prove a

theorem in which we relax the regularity of A, weaken property (3.1), and construct an unbounded

sequence that is summed by A.

Theorem 3.2. If A is a four dimensional summability matrix whose column sequences tend to

zero and

lim inf
k,l

{max
m,n

|amnkl}| = 0, (3)

then A sums an unbounded sequence.

Proof. Using (3.2), we choose increasing sequences of column indices (kp) and (lq) such that for

each p and q,

max
m,n

|amnkplq| < 2
−(p+q). (4)

Then choose increasing row indices (vp) and (wq) so that if k ≤ kp, l ≤ lq, m > vp and n > wq,

then |amnkl| < 2
−(p+q). Now define double sequence x by

xkl =

{
(p + 1)(q + 1), if k = kp, l = lq for some p, q

0, otherwise.



94 Fatih Nuray & Richard F. Patterson CUBO
17, 2 (2015)

Then m > vp and n > wq implies

|(Ax)m,n| = |

∞∑

i=0

∞∑

j=0

amnkiljxkilj|

≤

p∑

i=0

q∑

j=0

(i + 1)(j + 1)2−(p+q) +

∞∑

i=p+1

∞∑

j=q+1

(i + 1)(j + 1)2−(i+j)

= ((p + 1)(p + 2)(q + 1)(q + 2)2−(p+q)−2 + Rpq,

where limp,q Rpq = 0. Hence, limm,n(Ax)m,n = 0.

We note that if the row sequences of A tend to zero then (3.1) implies

lim
k,l→∞

{max
m,n

|amnkl}| = 0,

which stronger than (3.2). Therefore Theorem 3.2 does have a weaker hypothesis than Theorem

3.1.

Received: January 2014. Accepted: January 2015.

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[2] D. Djurcič, L. D. R. Kočinac, and M. R. Žižović, Double Sequences and Selections, Abstract

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[3] J. A. Fridy, Absolute summability matrices that are stronger than the identity mapping, Proc.

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[4] J. A. Fridy, Submatrices of summability matrices, Internat. J. Math. and Math. Sci., 1, (1978,)

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[5] H. J. Hamilton, Transformations of multiple sequences, Duke Math. Jour., 2 (1936), 29 - 60.

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17, 2 (2015)

Submatrices of Four Dimensional Summability Matrices 95

[9] R. F. Patterson, Four dimensional matrix characterization of absolute summability, Soochow

Journal of Math., 30 (1), (2004), 21-26.

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	 Introduction
	Main Results
	Summability of Unbounded Sequences