NONINCLUSION THEOREMS : SOME REMARKS ON A PAPER BY J . A . FRIDY

In 1997, J. A. Fridy gave conditions for noninclusion of ordinary and of absolute summability domains. In the present note, these conditions are interpreted in a natural topological context thus giving new proofs and also explaining why one of these conditions is too weak. Also an open question posed in Fridy’s paper is answered.

1. Noninclusion for ordinary summability.Recently, J. A. Fridy [2] stated a noninclusion theorem that can be formulated in the following way.Here lim n,k a nk = 0 (and, similarly, lim n,k b nk = 0) is taken in the Pringsheim sense, that is, ∀ > 0 ∃N > 0 : n > N and k > N ⇒ a nk < . ( Of course, this is a noninclusion theorem, since if A has that limit property and B does not, then c A c B .The reason for the above formulation is that it emphasizes an invariance property which is stated in an invariant form in the Lemma 1.2.Therein, e k denotes the basic sequence e k = (0,...,0, 1, 0,...) with "1" in the kth position, and the summability domain
As a corollary we obtain Fridy's result.

Corollary 1.3. Let A be a matrix with existing column limits and with row limits zero. If c
and then, in fact, B is a matrix with existing column limits and with row limits zero.
Proof.By the Lemma 1.2 we have e k → 0 in c A .By c A ⊂ c B , the relative topology of c B on c A is weaker than the FK-topology of c A (see [3,Ch. 17]; hence e k → 0 in c B , and, by Lemma 1.2, this means lim n,k b nk = 0, and the row limits of B are zero.
Remark 1.4.In [2] it is already noticed that in Theorem 1.1 the supposition that A and B should be regular can be relaxed to the condition that both matrices have column and row limits zero.Corollary 1.3 is slightly more general; the existence of the column limits of A is needed in order that e k ∈ c A for all k, and hence, by c A ⊂ c B , the column limits of B exist.It should also be remarked here that a K-space E containing ϕ is called a wedge space if e k → 0 in E, see G. Bennett [1, Thm.27], asserting that c A with ϕ ⊂ c A is a wedge space if and only if lim k→∞ sup n |a nk | = 0.

Noninclusion for absolute summability.
In [2] noninclusion is also considered for absolute summability; here the absolute summability domain of A, is concerned, where We state the result in the following form.
Theorem 2.1.Let A be a matrix with its column sequences in (so that e k ∈ A for all k), and let B be a matrix with A ⊂ B .If there is an index sequence (k(j)) j=1,2,... such that In [2], there is an extra condition ⊂ A , but condition (2. with the same µ as in (2.5).Unfortunately, this relaxed version fails for µ > 1, even if ⊂ A and the µ in (2.6) is allowed to differ from that one in (2.5).This can be seen from the following example.
To prove Theorem 2.1 in a topological way-similar to the proof of Corollary 1.3 (and Theorem 1.1)-we need the following lemma.Lemma 2.3.Let A be a matrix with its column sequences in , and let (k(j)) j=1,2,... be an index sequence.Then (2.10) Proof.The FK-topology of the FK-space A is given by the seminorms p r ,q r (see above) and (2.11) Thus e k(j) → 0 in A is equivalent to p r (e k(j) ) → 0,q r (e k(j) ) → 0 for each fixed r = 1, 2,... and Ae k(j) 1 = ∞ n=1 |a n,k(j) | → 0. These conditions are equivalent to the single condition Ae k(j) 1 → 0, since q r (e k(j) ) ≤ Ae k(j) 1 and p r H(e k(j) ) = 0 for k(j) > r .The lemma follows.
Theorem 2.1 is now a simple corollary of Lemma 2.3.By A ⊂ B the FK-topology of A is stronger than the relative FK-topology of B on A .Hence e k(j) → 0 in A implies e k(j) → 0 in B .Lemma 2.3 now yields the assertion of Theorem 2.1.
In [2] it is asked whether in Theorem 2.1 conditions (2.3) and (2.4) can be replaced by respectively.The answer is negative as can be seen by the following example.

Theorem 1 . 1 .
Let A and B be regular matrices such that c A , the summability domain of A, is included in c B , the summability domain of B. Then lim