ANALOGUES OF SOME FUNDAMENTAL THEOREMS OF SUMMABILITY THEORY

In 1911, Steinhaus presented the following theorem: if A is a regular matrix then there exists a sequence of 0’s and 1’s which is not A-summable. In 1943, R. C. Buck characterized convergent sequences as follows: a sequence x is convergent if and only if there exists a regular matrix A which sums every subsequence of x. In this paper, definitions for “subsequences of a double sequence” and “Pringsheim limit points” of a double sequence are introduced. In addition, multidimensional analogues of Steinhaus’ and Buck’s theorems are proved.


Introduction.
In [2,3,4,5,8], the 4-dimensional matrix transformation(Ax) m,n = ∞,∞ k,l=0,0 a m,n,k,l x k,l is studied extensively by Robison and Hamilton. Here we define new double sequence spaces and consider the behavior of 4-dimensional matrix transformations on our new spaces. Such a 4-dimensional matrix A is said to be RH-regular if it maps every bounded P-convergent sequence (defined below) into a P-convergent sequence with the same P-limit. In [9] Steinhaus proved the following theorem: if A is a regular matrix then there exists a sequence of 0's and 1's which is not A-summable. This implies that A cannot sum every bounded sequence. In this paper, we prove a theorem for double sequences and 4-dimensional RH-regular matrices that is analogous to Steinhaus' theorem. One of the fundamental facts of sequence analysis is that if a sequence is convergent to L, then all of its subsequences are convergent to L. In a similar manner, R. C. Buck [1] characterized convergent sequences by: a sequence x is convergent if and only if there exists a regular matrix A which sums every subsequence of x. We characterize P-convergent double sequences as follows: first, we prove that a double sequence x is P-convergent to L if all of its subsequences are Pconvergent to L; then we prove that a double sequence x is P-convergent if there exists an RH-regular matrix A which sums every subsequence of x. In addition, we provide definitions for "subsequences" and "Pringsheim limit points" of double sequences and for divergent double sequence.
x k,l − L < whenever k,l > N. We describe such an x more briefly as "P-convergent." Definition 2.2 (Pringsheim, 1900). A double sequence x is called definite divergent, if for every (arbitrarily large) G > 0 there exist two natural numbers n 1 and n 2 such that |x n,k | > G for n ≥ n 1 , k ≥ n 2 .
Definition 2.3. The sequence y is a subsequence of the double sequence x provided that there exist two increasing double index sequences {n i j } and {k i j } such that . . .
. . .   Figure 1 for an illustration of the procedure for selecting terms of a subsequence. A 2-dimensional matrix transformation is said to be regular if it maps every convergent sequence into a convergent sequence with the same limit. The Silverman-Toeplitz theorem [6] characterizes the regularity of 2-dimensional matrix transformations. In 1926, Robison presented a 4-dimensional analog of regularity for double sequences in which he added an additional assumption of boundedness. This assumption was made because a double sequence which is P-convergent is not necessarily bounded. The definition of the regularity for 4-dimensional matrices will be stated below, with the Robison-Hamilton characterization of the regularity of 4-dimensional matrices.
Definition 2.4. The 4-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. Remark 2.2. Definition 2.5 can also be stated as follows: a double sequence x is P-divergent provided that either x contains at least two subsequences with distinct finite limit points or x contains an unbounded subsequence. Also note that, if x contains an unbounded subsequence then x also contains a definite divergent subsequence.

Remark 2.3.
For an ordinary single-dimensional sequence, any sequence is a subsequence of itself. This, however, is not the case in the 2-dimensional plane, as illustrated by the following example. neither subsequence is x.
The following proposition is easily verified, and is worth stating since each singledimensional sequence is a subsequence of itself. However, this is not the case for double-dimensional sequences.
Proposition 2.1. The double sequence x is P-convergent to L if and only if every subsequence of x is P-convergent to L.
3. Main results. The next result is a "Steinhaus-type" theorem, so named because of its similarity to the Steinhaus theorem in [9] quoted in the introduction.

Theorem 3.1. If A is an RH-regular matrix, then there exists a bounded double sequence x consisting only of 0's and 1's which is not A-summable.
Proof. Let m i ,n j ,k i , and l j be increasing index sequences which we define as follows: Let whenever m 0 ,n 0 > B. Also, by RH 1 , RH 3 , RH 4 , and RH 5 we choose k 1 > k 0 and l 1 > l 0 such that Next use RH 1 , RH 2 , RH 3 , and RH 4 to choose m 1 > m 0 and n 1 > n 0 such that  (3.6) In general, having we choose m i > m i−1 and n j > n j−1 such that by RH 1 , (3.8) and by RH 3 , RH 4 .
In addition, by RH 2 . (3.11) We now choose k i+1 > k i and l j+1 > l j such that .
(3.12) Define x as follows: x k,l =    1, if k 2p < k < k 2p+1 and l 2t < l < l 2t+1 for p, t = 0, 1, 2,..., Let us label and partition (AX) m i ,n j as follows: k i <k<k i+1 ,l j <l<l j+1 a m i ,n j ,k,l x k,l , (3.14) where the general term a m i ,n j ,k,l x k,l is the same for each of the nine sums. Note that, , .
As with the original Steinhaus Theorem [9], we can state the following as an immediate consequence of Theorem 3.1.

Corollary 3.1. If A is an RH-regular matrix, then A cannot sum every bounded double sequence.
The next result is a "Buck-type" theorem. Proof. Since every subsequence of a P-convergent sequence x is bounded and P-convergent, and A is an RH-regular matrix, then for such an x there exists an RH-regular matrix A such that S {x} ⊆ C A .
Conversely, we use an adaptation of Buck's proof [1] to show that if A is any RH-regular matrix and x ∈ C then there exists a subsequence y ∈ S {x} such that Ay ∈ C . Note that every subsequence of x is bounded and x ∈ C , which implies that x has at least two distinct Pringsheim limit points, say α and β. Thus there exist increasing index sequences {n j } and {k i } such that lim sup x n i ,k i = α and lim inf x n i ,  Hence, {y * n,k } contains a subsequence {ȳ * n,k } with infinitely many 0's and 1's, along its diagonal. This implies that S {ȳ * } contains all sequences of 0's and 1's. Thus by Theorem 3.1, there existsỹ * ∈ S {ȳ * } such that Aỹ * ∈ C . Also, P-lim(y − y * ) i,j = 0. We now select a subsequence {ỹ i,j } of {y i,j } with terms satisfying lim sup i y n i ,k i = 1 and lim inf i y n i ,k i = 0 corresponding to the 0's and 1's, respectively of {ỹ * i,j }. Therefore P-lim(ỹ −ỹ * ) i,j = 0 andỹ i,j −ỹ * i,j is bounded. By the linearity and regularity of A, A(ỹ −ỹ * ) i,j = (Aỹ) i,j − (Aỹ * ) i,j and P-lim A(ỹ −ỹ * ) i,j = 0. Now since Aỹ * ∈ C , it follows that Aỹ ∈ C ; and sinceỹ =x − β/α − β, we have Ax ∈ C .