TAUBERIAN CONDITIONS FOR CONULL SPACES

The typical Tauberian theorem asserts that a particular summability method cannot map any divergent member of a given set of sequences into a convergent sequence. These sets of sequences are typically defined by an “order growth” or “gap” condition. We establish that any conull space contains a bounded divergent member of such a set; hence, such sets fail to generate Tauberian theorems for conull spaces.


I. INTRODUCTION
In this note we establish that a broad class of Tauberian conditions that hold for regular matrix methods cannot hold for conull methods.In particular, we consider "gap" and "order growth" conditions and show they are not Tauberian conditions for conull spaces.
Before proceeding with the discussion, we pause to collect some definitions and theorems.We let {the set of all sequences} {x : x is finitely nonzero} Recall that a locally convex Frechet space is FK-space if it is a vector subspace of and the coordinate functionals are contiuous.
We call an FK-space a sequence An FK-space is pre-conull if every semi-conservative space containing it is conul 1.
We list some well-known facts in the following two theorems.([2], [3]) Theorem-Let E F be FK-spaces with E F (set theoretically) and A a matrix map.Then (a) the inclusion map from E into F is continuous; Theorem: The intersection of two conull spaces is conull.
We also use a characterization of conservative conull spaces.Let r (r) be n be an increasin sequence of natural numbers with r I. Define Or(x) max{Ix u Xvl r u v n n rn+l and set (r) {x m: limit or(x) O} O r(x) for x e (r), then ((r), II'I r) is a con- If we define lxl Ir Ixll + SUPn n servative conu11 space.In fact, we also have from [3]: Theorem: Let E be conservative.E is conull if and only if E a (r) for some r.Now we are ready to begin.Let E be an FK-space and P We say that "P is a Tauberian condition for E" provided (*) x e P E implies x c.
Note that (*) is the general form of a (matrix summability) Tauberian theorem.The candidates for playing the part of P are defined as follows. Definition: ("gap" conditions) Let s (s) be an increasing sequence of natural ("order growth" conditions) Let (X) be a sequence of positive real numbers such that l .2. PRE-CONULL SPACES.
Our arguments hinge on two properties of pre-conull spaces: that they are neces- sarily "large" and that pre-conullity is preserved under intersection with a conull space.These properties are exposed in the next two lemmas.
Lemma i: If E is pre-conu[] and F is conu11, then E n F is pre-conull.
Proof: First observe that if H is a vector space containing E n F, then (E + H F) F H. This fo||ws from noting that if y E, z H F and y + z F, then y E F, and clearly E F + H F H.
Now suppose I is a semi-conservative space cntaining E F. Since F is n F II is semi-conservative (i.e., () is weakly Cauchy in F H), and consequently and since F is conull, (E + F H) n F is conull.Consequently H is also conull and we have established the lemma.
Iemma '2" If E is pre-conull, then (E .+ c (r) n for some r.In particular, E contains a bounded divergent sequence. Prod)f: It is easy to check that (E oo) + c (E + c) Since E + c is conserwtive, hence semi-conservative, and E is pre-conull, E + c is a conservative cnull space.Consequently E + c contains an f(r) for some r and (E ) + c !(r) .Now E contains a bounded divergent sequence since (r) does, i.e., (r) contains a bounded divergent sequence of the form y + z where y E and z c and y must be divergent (otherwise, y + z c).
We aIso give a sufficient condition for a space to be pre-conu11.
Lemma 3: E is pre-conull if there is a sequence (z n) a such that z converges to e in E and sup (Az)kl o.
k=l Proof: This foll(ws from the fact that semi-conservative space F is conul, if (nd only if) there is n sequence (z n) ; such that z converges to e in F and sup n k--'l 3. THE MAIN RESULT.
Lemma 4: If ('(s), O(X) and B(X) are defined as in section l, they are all pre-eonull spaces.
Proof: First we establish that C(s) is a pre-conull space.Observe that G(s) is a closed subspace of m when is given the topology of coordinatewise convergence (m's FK-topology), hence it is an FK-space.
Now let E be any semi-conservative space containing G(s) and set n k s ej o k= 1, 2, j=l n k Since (o cnveres to e in G(s), it converges to e in E. Now we have that (o n is weakly Cauchy in E and has a subsequence which converes weakly to e, hence (o n is weakly convergent to e in E. Thus F is conul], and consequently, G(s) is pre-conull.
nth unit vector, o e j=l and, if E is a Frechet space, E' denotes its continuous dual.
E is the FK-space of Caesaro summable sequences and we let (S n -I S satisfy limitn Sn+I/Sn and %n n then O(X) 0 and g(s)are Tauberian conditions for E. Also, J. Fridy has shown that for any real regular matrix A there is always an such that C(s) n [ is a Tauberian condition for the summa- bility field of A[4].
is weakly Cauchy, and (c) conull if (o n converges weakly to e.
space if it contains Definition: A sequence space E is (a) conservative if E-c, (b) semi-conservative if (o n