Internat. J. Math. & Math. Sci.
(1987) 199-202
199
Vol. i0 No.
RESEARCH NOTES
A NEW PROOF OF A THEOREM OF TOEPLITZ IN SUMMABILITY THEORY
P. ANTOSIK and V.K. SRINIVASAN
Department of Mathematics
University of Texas at E1 Paso
E1 Paso, Texas 79968
(Received January 24, 1985 and in revised form June II, 1985)
ABSTRACT.
The
object
of
paper
this
is
to
give
a
proof
new
of
a
Theorem
in
Summability, as an application of a result of Antosik.
KEY WORDS AND PHRASES.
Summability.
1980 AMS SUBJECT CLASSIFICATION CODE.
I.
40C.
INTRODUCTION.
From time to time as general concepts develop, new proofs of old theorems are
made
The
possible.
called
so
theorem
of
Toeplitz
in
complex numbers) numbers to be regular.
in Hardy
of
theory gives
summability
necessary and sufficient conditions for an infinite matrix
A
(a
of real (or
m,n
The first proof, entirely analytical is given
The second pr6of is obtained by using the uniform boundedness principle
[i].
Functional
In this paper in section 2,
analysis.
we
Toeplitz’s theorem based on a result of Antosik given in
obtain
his paper
a
new
[2].
of
proof
The above
method has other applications also.
2.
MAIN RESULTS.
Let
(a
A
be an
m,n
infinite matrix of real numbers.
Given a sequence
[s
n
of real numbers if
t
E
m
exists for each
a
L
L
n+
{s
We also say that
Given
denotes
(2.2)
is called the
A (lim s
(c)
(2.1)
n
and if
m
im+im tm
exists, then
s
m,n
n=l
n
A(lim s
n+
L
(2.3)
is A-summable to
n
and we write
n
L
if (2.3) holds.
(a
A
the
let A* be the linear space of all A-summable sequences. If
m,n
space of convergent sequences, then
is called a convergence
A
preserving transformation if
(c)
A*.
A
is called a regular transformation if
is convergence preserving and in addition for each convergent sequence
A (lira
n+=
Sn)
lira s
n+=
n
A
[s
(2 4)
200
P. ANTOSIK and V. K.
SRINIVASAN
We now state the diagonal theorem of Antoslk [2] used in this paper.
THEOREM A (Diagonal Theorem for Non-negatlve Matrices).
Let
A
(xl,
be a
matrix of non-negatlve real numbers such that
311 (xi, j)
0 (for all i
(xi, j)
0 (for all j
}+i (xi, i)
E
N)
(2.5a)
N)
(2.5b)
(2.6)
0
then there exists an increasing sequence of positive integers
E
N
such that
<
x
i,j
{pl}
Pi ’Pj
(2.7a)
being the set of natural numbers.
such that
I
i
x
0
and
PlPj
j=l
{pl}
Hence there exists
i
I
x
i=l
Pl
PJ
(2.7b)
0
We now state and prove the theorem of Toeplitz.
Let
THEOREM i.
conditions for
A
(a
m,n
A
be an infinte matrix.
The necessary and sufficient
to be regular are that the following three conditions must hold.
(b)
ml
(c)
ml
m,n
I
for each n.
0
a
I.
a
n=l
m, n
PROOF. We now prove the sufficiency part of the theorem.
Let A be an infinite matrix satisfying the conditions
(a), (b) and (c). Let
be a sequence such that s
s as n
Then
t
converges for each m.
n
m
now show that t
s as m
.
m
>
Given
.
0,
there is an integer
there is an integer M such that
a
m,n
For
m
>
M,
<
E
for
N
such that
m
>
M.
Is
s
n
<
for n
>
N.
{s
n
We will
By (c)
(2.8)
using (a) we have
m,n
n=l
n
N
n=l
,n
(2.9)
NEW PROOF OF A THEOREM OF TOEPLITZ IN SUMMABILITY THEORY
lam ,n
for
M’
<
M
such that
/N
(2.10)
sp(Is- Snl)
L
If
m.
<
>
M’
But by (b), there is an integer
201
we have for
>
m
M’
(2.11)
(2.11) shows that
t
as
s
=.
m
We now prove the necessity part.
e
(k)
{e
e
e
(k)
}
n
noting that
and
n
k
k
n
(2.12)
whee for all
n
(2.13)
I.
n, e n
The necessity of (b) in Theorem
belong to (c).
e
A
if
if
0
n
We first set
is obtained by
is convergence preserving and hence
In
(e
n
(k))
A
0
en(k))
(in
The necessity of condition (c) in Theorem
en
lira
n+=
A
is obtained by observing that
m=
en
(Inm
im am, k
i m
"
a
n=l
m,n
Thus we need to demonstrate the validity of condition (a).
Suppose (a) does not hold.
Then there exists a sequence of finite subsets
{o
m
of positive integers such that
<
max (o)
m
for
m
N (where
Let
Let
O
m
b
(2.14)
m+1
is the set of positive integers) with the additional property
N
Z
j
min (o
(2.15)
a
m,j
m
r.
je
o
a
m
-1/2
E
m’n
m,n
o
a
m
(2.16a)
for m e N
m,j
(2.16b)
m,j
We note that the columns and rows of the matrix
(b
converge to zero.
"m
=I
ires
I
bim
We now define a sequence
n
0
I)
i
(b
as follows.
[s
with
if
(See [2]
--0
n
i
n
U
Oik
if
k
Further it
m)n
is not difficult to show that there is a submatrix
i
m
’in
of
(b
m,n
diagonal theorem).
such that
(2.17)
Let
n
o
i
k
(2.18)
P. ANTOSIK and V. K. SRINIVASAN
202
{Sn
Obviously
(Co)’
the space of null sequences.
l b.
I
i
n=l
m n
By the regularity of
.
m
Hence
A
Z
j=l
a
(2.19)
sj
i ’j
m
j=l
it is seen that
a
j=l
ml
Z
ai m
Hence
b
exists
ira’ j sj
i
(2.20)
0.
m’ i n
(2.21)
Clearly we have
m’
for
m
nero
m’
(2.22)
n
n=l
n
N.
We use (2.17) and (2.21) to obtain
(Ibim ,i n
0.
(Ibim’in
On the other hand
(2.23)
I.
(2.24)
The apparent contradiction in steps (2.23) and (2.24) shows that condition (a) must be
valid. This now completes the proof of Theorem I.
REMARK I.
The Diagonal theorem is mainly used to show the necessity of condition
The remaining parts of the proof are quite standard and we presented them for
the sake of making the proof complete.
(a).
The authors would like to express their thanks to the referee for his suggestions
that resulted in the present form of this paper.
It is also possible to prove the
non-archimedean analogue of Toeplitz’s theorem, proved by Monna
using
the
analogue of
valued fields.
uniform boundedness
principle
over
[3], who derived
it by
complete non-archimedean
As the proof is a trivial modification of the same techniques used in
the classical case, this is left out.
REFERENCES
I.
2.
HARDY, G.H.
ANTOSIK, P.
Sequences
Divergent Series, Oxford University Press, London, 1949.
A
of
Diagonal
mappings,
Theorem for
Bull. Acad.
non-negative Matrices and Equicontinuous
Polon. Sci. Ser. Math. Astronom. Phys., 2__4
(1976), 855-859.
3.
MONNA, A.F.
Sur le Theoreme de
Wetensch., A 66, (1963), 121-131.
Banach-Steinhauss,
Proc.
Kon.
Ned. Akad. v.