NONINCLUSION THEOREMS FOR SUMMABILITY MATRICES

For both ordinary convergence and ℓ1-summability explicit sufficient conditions on a matrix have long been known that ensure that the summability method is strictly stronger than the identity map. The main results herein show that a matrix that satisfies those conditions can be included by another matrix only if the other matrix satisfies those same conditions.

I. INTRODUCTION AND TERMINOLOGY Let x denote a complex number sequence {xk}k__1 and A denote an infinite matrix [ok] with complex entries; then Ax is the transformed sequence whose n-th term is given by (Ax)n , cxk.k=l Let c denote the set of convergent sequences, and c. 4 {z Az E c} Similarly, 21 z" Ixl < and 2.4 {z Az E 21 The matrix ,4 is called regular if c C_ c.4, and ,4 is stronger than (ordirry) comergence if c c. 4 Similarly, ,4 is called an 2 2 matrix if 21 _ 2.4, and ,4 is stronger than the Mentty (map) if 21 2.4.If A and B are matrices such that limAz L implies limBz L, then we say "B includes .4.," and this clearly implies that c.4 C_ cB.In the 2-2 case we simply write 2.4 C_ 2B with no verbal phase describing it.
There is previous work giving explicit conditions on .,4 to imply that c.4 c c, or 2A 21 2 (See, e.g., the Mercerian-type theorems in [1], [3], and [41.In [2] and [5] conditions on ,4 were given that ensure that c c.4 and 21 2.4, respectively Explicit conditions are not known for making general comparisons of c.4 and c or of 2.4 and 2e (except when B I) In this paper we address the general inclusion question.The principal results show that if .4 satisfies the conditions of [2] or [5] that ensure that ,4 is stronger than I, then ,4 can be included by B only if B also satisfies those same conditions For the reader's convenience we state the theorems due to Silverman-Toeplitz [6, page 43] and Knopp-Lorentz [7] that characterize regular matrices and 2 2 matrices, respectively SIL%rEN-TEPLIT TItEOREM.The matrix A is regular if and only if the following conditions are satisfied: (i) for each k, lira , 0, (ii) for each n, o converges and lim o.,, 1, k=l k=l (iii) sup la.l < oo k=l 512 A. FRIDY KNOPP-LORENTZ TItEOREM.The matrix A is anmatrix if and only if the following condition is satisfied: (iv) sup lal < oo.k n=l 2. COMPARISON OF REGULAR MATRICES In [2] Agnew proved the following simple criterion for establishing c CA THEOREM 2.1.If A is regular and satisfies the condition lim ak 0, (2.1) then c cA.
The double limit in (2.1) is taken in the Pringsheim sense: if > 0 then there exists an N such that [ark[ < whenever both n > N and k > N.This sets the stage for the first of our "noninclusion theorems."THEOREM 2.2.If A and B are regular matrices such that A satisfies (2.1) and B does not, e., lim b, # 0, (2.2) PROOF.First note that since the rows of A are null sequences, (2.1) implies that l {mel l} =0.
Also, (2.2) allows us to choose increasing sequences d and W of row and column indices satisfying Ibv(r).'(,)l> 6 > 0 for all m. ( Then use (2.3) to choose a subsequence of those pairs < v" (m), n" (m) > such that max la.,,()l < 2-" for all m. ( Next, using conditions ST(i) we choose a further subsequence < v(m), (m) > so that for each m, k < (m)and n > v(m)imply lak[ < 2 and lb.kl < 2 m.
RET&RK.Tn the proof' of' Theorem 2.2 we did not use the ull strength of he regularity hypothesis.It would have sufficed to assume only hat the rows and columns of A and tend to zero.
To illustrate Theorem 2.2 we can take A to be any Cesro matrix C for j > 0, or any Euler-Knopp matrix for 0 < r < I. (They all satiffy (2.1).)Then could be any NOrlund matrix .N', with p finitely nonzero (s 6 page 6}), or any weighted mean Nwith p E (see r6, page 571), they satisi (2.2).Therefore none ofthe laer matris includes any of'the former.
One nght note the similarity of form bevvn Theorem 2.2 and Theorem 2.0.3 of" $] where Wilansky proved that if" A is conull and B is not, then c cB.The conservative matrix is conull provided that / -0.k=l

COMPARISON OF MATRICES
In [5] the following theorem was proved, giving a sufficient condition for anmatrix to be stronger than the identity matrix THEOREM 3.1.If A is an matrix for which there exists an integer m such that E I'k 0, (3.1) liminf then 1 A.
We next give an analogue of Theorem 2.2.THEOREM 3.2.If A and B arematrices such that A satisfies (3.1) and B does not, then Actually, we shall prove somewhat more.THEOREM 3.3.Let A be an matrix for which there is an integer/ and a sequence {k(3)}3= of column indices such that li3m E la,-,,k(3) O; then .4g B.
PROOF.First note that we may assume that the rows of B satisfy lira b..k(3) 0 for each n.
Assume that (3.2), (3.3), and (3.4) hold We shall find an x in gA that is not in gB.Using (3 3) and replacing { k(j)} with one of its (appropriately chosen) subsequences { k(i)}, we can assume without loss of generality that ' Ib.<ol > 25 > 0 for each i.
(3.6) n--1 Replacing { k(i) with yet another of its subsequences { k(p) } we can get E la",(n) < t, for each p, ( where Next we construct an increasing sequence {z,,(m)} of row indices and a further subsequence (m)} of {k(p)} to define the sequence z that we seek First, take u( 1) 0; then use (3 6) to choose (1) m=l m 3= Hence, Bx gl, which establishes the assertion that x is in gA but not in gB.
Note that in defining x we need only have Iz(.)l_< lira in order to have the subsequent inequalities valid It is the convergence of the tz 1 series a.,:O)z,,O)

3=1
for n 1,2, ...,# 1 that requires the factor ofe ' in x(m) (See Theorem and Lemma of [5] REMARK.As above with Theorem 1, we have not needed the full strength of the hypotheses, in this case, the assumption that A is an t-matrix is stronger than what is needed.Condition (3 2)   guarantees that Ax ga whenever it exists, so the only concern is that (Ax),,, exists for n < # This existence would be guaranteed by assuming only that the row sequences {a.k0)}a=l for n </ are bounded.(See Lemma of [5].) k n(rn), for m 1, 2, and 0 E R otherwise.This yields z E gA because by(3