I nternat. J. Math. & Math. Si.
503
Vol. 6 No. 3 (1983) 503-510
INFINITE MATRICES AND ABSOLUTE ALMOST CONVERGENCE
MURSALEEN
Department of Mathematics
Aligarh Muslim University
202001, INDIA
Aligarh
(Received January 26, 1981 and in revised form August 16, 1982)
ABSTRACT.
In 1973, Stieglitz [5] introduced a notion of
FB-Convergence
which provided
convergence due to Lorentz [i].
a wide generalization of the classical idea of almost
The concept of strong almost convergence was introduced by Maddox [3] who later on
generalized this concept analogous to Stieglitz’s extension of almost convergence [4].
In the present paper we define absolute
the concept of
FB-Convergence
which naturally emerges from
FB-Convergence.
KEY WORDS AND PHRASES. In fine matces, almost convergence, stng almost convergence, FB-Convergence, absolute FB-Convergence.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
1
40C05, 40D05.
INTRODUCT ION
Let
o’
c, and c o denote respectively the Banach spaces of bounded, convergent,
(Xk)
and null sequences x
of complex numbers with norm
lxll
sp Xkl
and let v
be the space of sequences of bounded variation, that is,
v
Suppose that B
Then x
oo is
{x:
(B.)
said to be
lim
n
uniformly for i
The space F
B
(Bi).
llxll-
k=O
Ixk Xk_ll
<
0}.
+ oo, x_ I
is a sequence of infinite complex matrices with
FB-Convergent
[5],
lira
n
p=O
(Bix) n
to the value Lim
bnp(i)xP
Bi=(bnp(i)
Bx, if
Lira Bx,
0,I,2
of
B
In case B
FB-Convergent
B
0
sequences depends on the fixed chosen sequence
(I) (unit matrix), the space F B is same as c and, for
504
B
MURSALEEN
B
B(1).
i
(I)
(B i ), it is same as the space f of almost convergent sequences [i], where
I
(b(1)
np
(i)) with
I
b(l)
(i)
np
I
i
n+l
!
0
-< p -< i + n
elsewhere
Maddox [4] generalized strong almost convergence by saying that x
and only if
_Dnp(i)
IXp
P
0 (n- oo, uniformly in i)
s
-
P
s[F B] if
(i.1)
assuming that the series in (i.i) converges for each n and i.
In particular, if B
B0,
the
[FB]
c; if B
strongly almost convergent sequences [3].
entry), 0
(i,I,i
and e
B i, then
We shall write e
[FB]
[f], the space of
(0,0,...,0,i (kth
k
).
Let s be the space of all complex sequences and
d
F
B
B
{x
s:
{x
(dB ):
Lim Bx
lira
exists for each i}
(BiX)n
lim t (i x) exists uniformly in i
n
and the limit is independent of
i},
where
g bnp(i)x
t
n
p
p0 0p(i)Xp
(i,x)
(n >- I)
(n
0)
(n =-I)
0
and
0
otherwise.
Let
n
(i x)
x)
t (i
n
tn-i (i,x)
Therefore, we have
[bnp(i)
n
bn_l, p (i)]Xp,
(n -> l)
(1.2)
(i,x)
Op(i)
x
P
(n
0)
INFINITE MATRICES AND ABSOLUTE ALMOST CONVERGENCE
DEFINITION.
A sequence x
Let B
(Bi)
be a sequence of infinite matrices with B
FB-Convergent
is said to be absolutely
19
if
(i x)
505
i
(bnp(i)).
converges
n
n=0
uniformly for i >- 0, and lim t (i,x) which must exist should take the same value for
n->oo n
all i.
2.
We denote the space of absolute
FB-Convergent
sequences by v(B).
THE MAIN RESULT.
In this note, we denote by (v,v(B)) the set of matrices which give new classes
of absolute B-conservative matrices and absolute almost B-conservative matrices.
Let A be any infinite complex matrix for which the pth row-sum converges for a
given x for all x in some class.
We have
A x
P
(Ax)
apk
P
x
k
and
p0 bnp(i) Xp
(BiX)n
Therefore,
(BiAX)n
bnp(i)
p=o
np
A
P
k=O
apk xk"
and, assuming the interchange of order of summation can be justified (see lemma), we
get that
(BiAx) n
k=O
p--0
bnp (i) apk
(2.1)
x
k
Now, by (1.2) and (2.1), we have
n(i’Ax)
tn(i,Ax)
tn_ l(i,Ax)
p0 [bnp(i)- bn-l’
p
(i)
]ApX,
(n
(n
k=O
where
gnk (i) Xk,
.
I),
0),
(2.2)
506
MURSALEEN
p=O
gnk(i)
p=O
HEOREM.
SUPn
Let A be an infinite matrix.
sup
p,k
30p (i)
>
(n
apk
p0=
Then A:
me
=k
< o,
bnp (i)
i),
0).
for each i.
v--> v(B) if and only if
<
there is an N such that for r,i
(ii)
(n
bn_l, p (i)]apk
(B i) be a sequence of infinite matrices with
Let B
(i)
[bnp(i)
0,i,2,...
r
E gnk(i)
n=N
(apk)p>_0
(iii)
(iv)
k=0
Let A e (v,v(B)).
<
e
v(B) for each k,
For each k, let
(Xk)
and
v(B)
apk) p->0
apk
be
FB-Convergent
with limit
k"
And let
(In each case, limit is taken for p > 0).
k=OE apk be FB-Convergent with limit c.
If x
(constant),
K
k=0
v, then
lim t (i Ax)
lira x
k
k-o
n-o n
+
k=0
(xk
k) k"
lim x
k-o
We use the following lemma in the proof.
LEMMA.
If either the necessity part or the sufficiency part of the theorem
holds, then, for x
v,
Xk p0 bnp(i)
p=0 bnp(i) K=0 apkxk kOapk’=
PROOF.
If either A:
v-
v(B) or the conditions (i)-(iv) of the theorem hold,
then by part_ i summation, for x
k=O
where
dpk
a
%=k
p%"
v,
apkXk
k:O dpk
(xk
Since condition (i) holds,
dpk
Xk_ I)
is bounded for all p,k.
Thus
.
p=O
INFINITE MATRICES AND ABSOLUTE ALMOST CONVERGENCE
bnp (i)
apkxk
d
bnp (i)
p=O
k=0 (xk
=
k=O
Xk-l)
pk
(x
507
Xk-i
k
bnp(i) ape’
(where the inversion is justified by absolute convergence)
bnp(i) apk
xk
k=0
since
l im x
k_o
PROOF OF THEOREM.
Since
k
p0
b
np
(i) d
0.
pk
Condition (i) follows from the fact that A: v
Necessity.
v, necessity of (iii) and (iv) is obvious.
ek,e
It is clear that, for fixed p and j,
J
x
/
k=0
apk
x
k
We are given that, for all x e v, it tends to
is a continuous linear functional on v.
(for fixed p) and hence, by the Banach-Steinhaus Theorem [2], this
a limit as j
limit A x is also a continuous linear functional on v.
P
We observe that, although
Y.
n=0
ln(i,Ax)
for r
i and
>- o,
{qN,i
i
>- o,
qm, i
n=l
is pointwise bounded on v.
i_> o
Such an N exists.
For suppose not.
qr i(Xr)
By the principle of condensation of singularities [6],
v:
sup
i_>o
qr
o but not uniformly
v, and there is an integer N such
v with
sup
e
o
on the behaviour of
i_>O
{x
->
(i,Ax)
[n(i’Ax)["
is a continuous seminorm on
0,1,2,... ther exists x r
n
it needs
Thus, by definition, there is an m such that
qm,i(x)
that
(i,Ax)
is uniformly convergent in i
(i,Ax) for sufficiently large n.
For m
t
Now, we can say that uniforn convergence bears only
bounded.
n
l
n-o
is uniformly convergent in i
For example, if
not be uniformly bounded in i.
(n -> i and i), then
In (i,Ax)
i
(x)
for
r
0,1,2,..
Then
508
MURSALEEN
is of second category in v and hence nonempty, i.e., there is x
sup q
i>o
r, i
(x)
for
v with
0,1,2
r
But this contradicts the fact that to each x e v there exists an integer N
i->oSUp qNx,i(x
x
with
< o.
Now, by another application of the Banach-Steinhaus Theorem, there exists
a con-
stant M such that
-< M
qN,i(x)
Apply (2.3) with x
(xk) defined by x
Ixll.
i for k
k
-< r and o for k
(2.3)
> r.
Hence (ii)
must hoid.
Sufficiency.
Suppose that the conditions (i)-(iv) hold and that x
defined v(B) as a subspace of %
Thus, in order to show that Ax
necessary to prove that Ax is bounded.
v(B),
v.
We have
it is
By virtue of condition (i), this follows im-
mediately.
Now,
it follows from (iv) and the lemma that
gnk (i)
k=O
converges for all i,n.
Hence, if we write
hnk(i)
then
hnk(i)
%--k
gnu(i),
is defined, also for fixed i,n,
hnk (i) --> 0
as k--.
(2.4)
Now condition (iv) gives us that
(2.5)
n=O
converges aniformly in i, and, for suitable chosen N,
n=N
is bounded.
lhno (i)
(2.6)
By virtue of condition (iii), for fixed k, we get that
n=O
converges uniformly in i.
Since
k-i
hnk(i)
hn0(i)
%=0
gnu(i)’
(2.7)
INFINITE MATRICES AND ABSOLUTE ALMOST CONVERGENCE
509
it follows that, for fixed k,
n=O
lhnk (i)
(2.8)
converges uniformly in i.
Now
(i,Ax)
gnk(i)
k=O
x
k
[hnk(i)
k=O
hnk(i)
k=O
h
(xk
n
k+l(i)]
x
k
(2.9)
Xk_l)
by (2.4) and the boundedness of x
k.
Condition (ii) and the boundedness of (2.6) show that
n=N
is bounded for all k,i.
lhnk(i)
(2.10)
We can make
k=k +1
o
arbitrarily small by choosing k
c>
o, we can choose k
o
o
sufficiently large.
It therefore follows that, given
so that, for all i,
k=k +1
n=N
hnk(i)(xk
<
Xk_l)
o
-
(2.11)
By the uniform convergence of (2.8), it follows that, once k has been chosen, we can
o
choose n
o
so that, for all i,
k
n=n +i
o
o
k=o
hnk(i)(xk
< g.
Xk_l)
It follows from (2.11) that the same inequality holds when
7. is replaced by
n=N
7.
n=no+l
hence, for all i,
n=n +1
o
hnk(i)(xk
k=O
Xk_l)
Hence,
n=n +i
o
ln(i,Ax)
Thus
n=O
I@ n(i,Ax)
< 2g.
< 2g.
(2.12)
510
_
MURSALEEN
converges uniformly.
Now, by virtue of (2.9), we have
lim
n-oo
tn(i,Ax)
-
tN_l(i,Ax)
__-- hnk(i
(x
k=0
N= hnk(i)
Xk-l)
(xk
Xk_l)
k
n=N k=O
(2.13)
the assertion being justified by absolute convergence because of the boundedness of
(2.10).
By (2.7), we have
n=N
Thus,
k-i
hnk(i)=
lim t
(i
n-o n
n=N
hno(i)=0
Ax)
%
lira x
k
k-o
%=0
%=k p=O
+
>-
k=0
gn%(i)
n=N
(xk
bN_ 1
p
lim
k-o
(i)
ap
"
Xk) k
This completes the proof.
REFERENCES
i.
LORENTZ, G.G.
A contribution to the theory of divergent series, Acta Math. 80
(1948), 167-190.
Press, 1970.
2.
MADDOX, l.J.
Elements of Functional Analysis, Cambridge University
3.
MADDOX, l.J.
61-64.
A new type of convergence, Math. Proc. Camb. Phil. Soc. 83 (1978),
4.
IviADDOX, l.J.
On strong almost convergence, Math. Proc. Camb. Phil. Soc. 85 (1979),
345-350.
5.
STIEGLITZ, M.
Eine Verallgemeinerung des Begriffs der Fastkonvergenz, Math.
Japon. 18 (1973), 53-70.
6.
YOSIDA, K.
Functional Analys__i__s, Springer-Verlag, Berlin-Heidelberg-New York,
196