Internat. J. Math. & Math. Sci.
Vol. 8 No. 4 (1985) 689-692
689
TAUBERIAN CONDITIONS FOR CONULL SPACES
J. CONNOR
Department of Mathematical Sciences
Kent State University
Kent, Ohio 44242 U.S.A.
and
A. K. SNYDER
Department of Mathematics
Lehigh University
Bethlehem, PA 18015 U.S.A.
(Received February 15, 1984)
ABSTRACT.
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 s,quences are typically defined by an "order growth" or
We establish th,lt
aly
"gap" condition.
conu11 space contains a bounded divergent member of such a set;
hence, such sets fail to generate Tauberian theorems for conull spaces.
YaEY WORDS AND PHRASES. FK pace; conull.
1980 ATHEMATICS SUBJECT CLASSIFICATION CODE.
I.
46A45, 40E05, 40H05.
INTRODUCTION
In this note we establish that a broad class of Tauberian conditions that hold
for regular matrix methods cannot hold for conull methods.
"gap"
In particular, we consider
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}
{x
:
limit x
n n
{x
:
limit x
=x
:
SUPnlXnl
c
o
c
exists}
}
(1,1,,..)
e
e
n n
0}
n
the
nth unit vector,
o
n
e
j=l
and, if
E
is a Frechet space,
E’
denotes its continuous dual.
Recall that a locally convex Frechet space is FK-space if it is a vector subspace
of
and the coordinate functionals are contiuous.
space if it contains
We call an
FK-space a sequence
J. CONNOR AND A.K. SNYDER
690
A sequence space
Definition:
if (o
n
E
(a) conservative if
is
(c) conull if (o
is weakly Cauchy, and
n
converges weakly to
Clealy conull spaces are semi-conservative.
E contains
c
if and only if
o
I
j=l
spaces are also semi-conservative.
if (e )I
E- c, (b) semi-conservative
e.
It is known that a sequence space
for all
f e
E’[IJ, hence
conservative
An FK-space is pre-conull if every semi-conservative space containing it
;)efinition:
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;
{x e :
(b) E A
Axe E} is an FK-space; (c) if E is conull, F is conull; and (d) if
E, E is
F
closed in
Theorem:
F
and
E
is conull, then
is conull.
The intersection of two conull spaces is conull.
We also use a characterization of conservative conull spaces.
be an
increasin sequence
r(x)
max{Ix u
(r)
{x
O
I.
of natural numbers with r
n
Xvl
r
limit
or(x)
n
v
u
Let
(r)
r
n
be
Define
rn+l
and set
If we define
lxl Ir
Ixll
servative conu11 space.
Theorem:
Let
E
+
m:
SUPn Onr(x)
E"
is a Tauberian condition for
x e P
for x e (r), then ((r),
II’I r)
is a con-
In fact, we also have from [3]:
be conservative.
Now we are ready to begin.
(*)
O}
E implies x
E
Let
is conull if and only if
E be an FK-space and
P
E a (r)
for some
r.
We say that "P
provided
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
n
numbers" set
G(s)
Definition:
{x
m:
(kx)
k
Recall that if
n
satisfy
E
limitn
Tauberian conditions for
A
Xk+
there is always an
bility field of A[4].
0 only if k
s
n
for some n e N}.
(X) be a sequence of positive real
Let
.
l
n=
6()
(S
k
("order growth" conditions)
numbers such that
S
x
(AX)n[
{x e :
o(k )}
n
FK-space of Caesaro summable sequences and we let
-I
and
Sn+I/Sn
are
%n n then O(X) 0 and g(s)
is the
E.
Also, J.
Fridy has shown that for any real regular matrix
such that C(s) n
[
is a Tauberian condition for the summa-
691
TAUBERIAN CONDITIONS FOR CONULL SPACES
PRE-CONULL SPACES.
2.
Our arguments hinge on two properties of pre-conull spaces: that they are necessarily "large" and that pre-conullity is preserved under intersection with a conull
These properties are exposed in the next two lemmas.
space.
If E is pre-conu[] and F is conu11, then E n F is pre-conull.
Lemma i:
H.
F
y
F, then (E + H
First observe that if H is a vector space containing E n
Proof:
This
fo||ws
F + H
H
E, z
from noting that if y
F, and clearly E
E
F and y + z
F)
F, then
H.
F
Now suppose I is a semi-conservative space cntaining E
F.
Since F is
n
H), and consequently
II is semi-conservative (i.e., () is weakly Cauchy in F
E + F n H is conull,
that
n
F
H
implies
E
+
E
Nw
semi-cnservative.
is
E + F
II
H is also conull and
Consequently
H) n F is conull.
and since F is conull, (E + F
F
.
we have established the lemma.
If E is pre-conull, then (E
Iemma ’2"
+
for some r.
(r) n
c
In particular,
E contains a bounded divergent sequence.
Prod)f:
Since E
)
+
.
+
c
(E + c)
+
c is conserwtive, hence semi-conservative, and E is pre-conull, E
+
is a conservative cnull space.
(E
oo)
It is easy to check that (E
!(r)
c
Consequently E +
c
c contains an f(r) for some r and
Now E contains a bounded divergent sequence since (r) does, i.e., (r) contains
a bounded divergent sequence of the form y
be divergent (otherwise, y
+ z
+
z where y
and z
E
c and y must
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)
in E and sup
k=l
Proof:
(Az)kl
converges to e
o.
This foll(ws from the fact that semi-conservative space F is conul,
only if) there is n sequence (z n)
3.
a such that z
;
such that z
if (nd
converges to e in F and sup n
k--’l
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
is given the topology of coordinatewise convergence
closed subspace of m when
(m’s
FK-topology), hence it is an FK-space.
Now let E be any semi-conservative space containing G(s) and set
n
o
k
s
ej
k= 1, 2,
j=l
n
Since (o
k
cnveres
to e in
G(s), it converges to e in E.
is weakly Cauchy in E and has a subsequence which
weakly convergent to e in E.
converes
Now we have that (o
weakly to e, hence (o
n
n
is
Thus F is conul], and consequently, G(s) is pre-conull.
J. CONNOR AND A.K. SNYDER
692
We now turn our attention to 5(%) and
bl,1
|’
B and
5(X)
fined by
()
v, y
te orm
that, since 5()
is
I
bn,n-I
-I’
_-I
n-I
n,n
(b
Consider the natrix B
Observe that
0 otherwise.
bn, k
and
de-
n,k
hence both are FK-spaces with the metric topology bein
(Co) B
If II
! )(),
b
(%)
maxIxI, u ]l<AxI
it suffices to show that
5()
Ao ot
’ ot c.
is
pre-conull
()
to show that
pre-conu11.
of Lemma 3.
We now construct a sequence in 5() that satisfies the hypothesis
-I
for all k e
Bk
be a sequence of reals such that 0
(ilk)
l.et
k-
max(k:
Z
i=n
fr k
-< n,
z
k-I
n
l
i=n
k
e
that
It’s
I}.
Bi
gi
IB
als clear that (z n)
_
for n
-I
max{ll k
for all n e N.
l(n)
Note that n
k -< l(n) and
kl:
n
k=l
0 for k
l(n)}, hence
<- k
Define z
n
z
n
n
by z k
It’s easy
l(n).
limitn
and set
to
e in 5().
for all n e N, hence, by Lemma 3,
l(AZn)k
and
z
and
k
125])
0 (this is possible by one of Abel’s results [6, p.
limitk %k fik
l(n)
N,k=l B
5(,) is pre-conull.
We can now show that neither 5(),
()
nor G(s) can generate Tauberian conditions
for a conull space.
Theorem:
If E is a conull space, then 5()
a bounded
diverent sequence.
E,
()
E and G(s)
E, 5()
5() and G(s) are all pre-conull, ()
sequence.
bounded
a
diverent
nre all pre-conu|l, hence each contain
Proof:
Since
(),
E all contain
E and G(s)
E
REFERENCES
BENNETT, G
and KALTON, N
J
"FK spaces containin
c
,"
Duke Math Jour., 39(3)
(1972), 561-582.
2.
3.
4.
5.
6.
"Summability through functional analysis," North-Holland, 1984.
SNYDER, A. K. "Conull and coregular FK spaces," Math. Zeitschr., 90(1965), 378-381.
FRIDY, J. "Tauberian theorems via block dominated matrices," Pacific Journal of
Math. 81(1979), 81-91.
SNYDER, A. K. and WILANSKY, A. "Inclusion theorems and semi-conservative FK
spaces," Rocky Mountain J. of Math. 2(4) (1972). 595-603.
KNOPP, K. Infinite Sequences and Series, Dover, 1956.
WILANSKY, A.