Internat. J. Math. & Math. Sci.
VOL. 21 NO. 4 (1998) 701-706
701
DIFFERENCE SEQUENCE SPACES
A.K. GAUR
Depmment of Mathematics
Duquesne University
Pittsburgh, PA 15282, U.S.A.
and
MURSALEEN*
Depanmem of Mathematics
Aligarh Muslim University
Aligarh 202002, INDIA
Cgcceived April 17, 1996 and in revised form July 29, 1996)
ABSTRACT. In
s,(a).
(:,,) (:’IA,I),%
{
co}
for r > 1 is studied. In this paper, we generalize this space to S,. (p, A) for a sequence of strictly positive
reals. We give a characterization ofthe matrix classes (S,(p, A),oo) and (S,.(p, A), 1).
KEY WORDS AND PHRASES:
Difference sequence spaces, KOthe-Toeplitz duals, matrix
transformations.
1991 AMS SUBJECT CISSIFICATION CODES: 40H05, 46A45.
INTRODUCTION
Let too, c and co be the sets of all bounded, convergent and null squences of
respectively. Let w denote the set of all complex sequences and 1 denote the set of all convergem and
absolutely convergent series.
Let z be any sequence and Y be any subset of w. Then
1.
Z -1"
Y
{z 6 u" zz
(akz)F 6 Y}.
For any subset X of w, the sets
X=
N (z-
e)
and X/
N
zEX
zEX
are called the a- and/-duals of X.
We define the linear operators A, A -] v --, v by
Ax
(Axe)?’
A-lz
A-12:k
(x x+)’,
and
such that
Z3
A.K. OAUR AND MURSALEEN
702
A-l:rl
O.
Let
In this paper we extend the space ,5, (A) to ,5 (p, A) in the same manner as co, c, oo were extended
co(v), c(.v), eoo(p), respectively (cf. [2],[3],[4]). We also determine the a- and/-duals of our new
sequence space. Let p (Pk) be an arbitrary sequence of positive reals and r >_ 1, then we define
to
whe
co(p)’=
If p
e
(1, 1, 1
{x
e v- limk...,oo [x[
=0}.
), then the set S (p, A) reduces to the set S (A). For r
O,
,sr (P, A) is the same
as Aco(p) (cf. [5],[6],[71).
We will ncd the following lemmas:
LEMMA 1 (Corollary in [7]). Let
,a
e (P,)-. ce implies R
(p)oo__ be a sequence of nondccreasing positive reals.
1,2, ...).
(P) e (P,)- co where P
ak (n
Then
k=n+l
LEMMA 2 (Lemma l(b) in [8]). Let p
Then A (co (p),
if and only if
N
for some integer M
2.
nite
(pk),= be a strictly positive sequence such that p
k=
> 2.
Tim a- AV B-DUALS O1 S, (, A)
TIIEOREM 2.1. Let p (Pk)’ be a strictly positive sequence and r
1. Then
-<0’),
[s(, A)]o
<)
>
N>I
(b)
vC
N>I
where
and
in co.
c is the set of all positive sequencesThen
PROOF. (a) Let a e
U D(I)(P)
a-s(1/No) 6 el for some No _> 2,
(2.1)
703
DIFFERENCE SEQUENCE SPACES
where
k=l
Since,
\j=
() is increasing, (2.1) implies that
aE
,
Let x e Sr(p,A).
suplkrAzk] pk <
k>_M
No 6 N,
Then for a given
and hence
--< k-
[Axk[
1.
(2.2)
there exists an M
for all k
1,2,
M(No). N such that
and consequently by (2.1) we have
(2.3)
k=l
k=l
3=1
Finally, by (2.2) and (2.3), we get
[S(p, zx)]
aE
Let a
q U D(I)(P)
.
Then we can determine a strictly increasing sequence
(k(m))m=,
of integers such
N>I
that k(1)
1 and
k(m+1)-I
I1(/(, + 1)) > 1 (rn
1,2,...).
=k(,,,)
(xk) by
We define the sequence z
min{k-l,k(i+l)-l}
=
Then x E
(i d- 1) -1/ps
(k(m) _< k _< k(rn + 1)- 1;rn
1,2,...).
=(k(,))
Sr (p, A) and
k(m+l)-I
k=l
m=l
k=k(m)
which proves that
[s,(, A)]
Hence, [Sr(p, A)]
.
U D(*)(P)"
n>l
_
(b) Let a E Or (p). Then a E c8, and Abel’s summation by parts yields
(kZk
k=l
Z RkAZk
AZk + x
@ tn
k=l
k=l
ak
for all x, (n
k=l
Further
-
r
’/
1
Let z E Sr (P, A). Then there is a sequence v E
vllw (
IAzI
_<
for all sufficiently large k. Now, by (2.5)
for some integer
No
2.
C such that
N; l/pk
1,2,...) and IAx, _<
1, 2, ...).
(2.4)
(2.5)
A.K. GAUR AND I$ALEEN
704
Hence
RAz
Finally, by Lemma 1, a
(Aflvl/’)
-1
(2.6)
cs implies that
(A’lvl/p)-I
J:{
(2.7)
C
and consquently
n-1
k=l
From a
cs, (2.4), (2.6) and (2.8), we conclude that
.
.
RkAZk + :r.1
akZk
k=l
k=l
(2.9)
ak
k=l
ca. Thus a [Sr(p, A)] Now, let a [S,(p, A)] Then az cs for all z S,(p, A) and
be given. Then z A71v 1/’ Sr(p,A). Hence
Sr(p,A). This implies that a cs. Let v
a (A71vl/)-I ca, and by Lemma 1, we get (2.7). Therefore (2.8) holds for all z Sr(p,A). By
co(), this
(2.4), we get RAx cs. Since z S,(p,A) if and only ifv A,z (krAzk)
and az
’e
c
=
implies that
for some integer N
> 2 (cf. [9], Theorem 6). Hence [,.%(p, A)]
C’r(p).
MATRIX TRANSFORMATIONS
For any infinite complex matrix A ak ),.k= 1, we write A.
ak k= for the sequence in the nth
row of A. Let X and Y be two subsets of w. By (X, Y), we denote the class of all matrices A such
a,kzk converges for all x X and each n N, and the sequence
that the series A,(z)=
3.
k=l
Az
A
Y for all x X.
THEOREM 3.1. Let p= (pk)
(A,(x)),=
be a strictly positive sequnce and r _> 1.
(s,0, A),eoo) tfand only if
=sup
EOkE
3=1
k-’l
Dr(M)"
(ii)
where k
k
supn
(k=l
IPk
M1/pr ]
<OO
for 1 n d k, d
k+l
(fii)
forsomimger M2,
D"
p JAn(e)]
p
k
[k=l
<
.
Then
705
DIFFERENCE SEQUENCE SPACES
PROOF. Let the conditions (i), (ii) and (iii) be true and x e Sr(p,A). By Theorem 2.1(b),
conditions (i) and (ii) imply that A, [Sr(p, A)]a for n 1,2,.... for a given M N, there exists a
M’ M’(M) N such that sup IkrAxl <_ where M _> 2 is the integer in (ii). By (2.9), we have
-,
k>_M’
IA,(z)]
D,.(M)+ IxlD (n
1,2
oo.
and hence Ax
Conversely, let A (S,.(p,A),oo). Since Z AI)1/p ’r(p,A) for all
v
condition (i) follows immediately. Also the necessity of (iii) follows from the fact that
x e S (p, A). Now, by (i), (iii) and (2.9),
c,
_,
(:)
Rk + A,,(e) (,
,2, ...).
k=l
Since Ax
Coo
ZlAe Coo, therefore (RnAz)n%l Coo. Since z S(p,A) if and only if
co(p), and
(Rnk/kr)(krAxk)
too for all (k Ax),=l co(p), this implies
(
(k"Ax,)=
that D,.(M)
A
and
k=l
< x for some integer M _> 2, and (ii) holds.
TIIEOREM 3.2. Let p (Pk) be a strictly positive sequence such that p
(S (p, A), g if and only if
CI) ()
(i)
sup
N N
too, and r _>
1. Then
A. (a;d/’)
k-1
sup
n.N k=l
vl/p
j=l
for all sequences v
(ii)
for some integer M _> 2, and
(iii)
D(3)"
vSUPN
i A,(e)
<oo.
N
M
.
PROOF. Let conditions (i), (ii) and (iii) hold. Then A,, [S(p, A)]a. Let xS,.(p, A). For a iven
N there exists a M’= M’(M) lI such that sup IkAxl pk <
Now, by (2.9) and the
k>M’
inequality in 10], p. 33, we have
< 4(C2)(M)4-IXllD3)) < oo.
la.(z)l
r--1
Since m
N is arbitrary, we have Az
nN
el. Conversely, let A (qr(p, A),el). Then
An(x’)
E]An(:F’)]k=I
< O0
for all z S (p, A) and for all finite subsets N of N. Therefore the necessity of (iii) and (i) follows
Further we have
immediately, since e and x A7lv l/r" S,.(p, A) for every sequence v
c.
A. IC GAUR AND MURSALEEN
706
and hence (ii) holds by a 2.
ACKNOWLEDGMENT. (,) This research is supported by the University Grant Commission, number
F.8-14/94. The authors are grateful to the referee for his or her valuable suggestions which improved the
clarity of this presentation.
CHOUDHARY, B. and MISHRA, S.K., A note on certain sequence spaces, J Analysis, 1 (1993),
139-148.
[2] LASCARIDES, C.G., A study of certain sequence spaces and a generalization of a theorem of Iyer,
Pacific J. Math. 38 (2) (1971), 481-500.
[3] LASCARIDES, C.G. and MADDOX, I.J., Matrix transformations between some classes of
sequences, Proc. Cambridge Phil. Soc. 68 (1970), 99-104.
[4] SIMONS, S., The sequence spaces g(p,) and m(p,), Proc. London Math. Soc. 15 (1965), 422436.
[5] AHMAD, Z.U. and MURSALEEN, KOthe-Toeplitz duals of some new sequence spaces and their
matrix maps, Publ. Inst. Math. (Beograd) 42 (56) (1987), 57-61.
H., On certain sequence spaces, CargutianMath. Bull. 24 (1981), 169-175.
KIZMAZ,
[6]
[7] MALKOWSKY, E., A note on the K6the-Toeplitz duals of generalized sets of bounded and
convergem difference sequences, J Analysis 3 (1995).
[8] MALKOWSKY, E., MURSALEEN and QAMARUDDIN, Generalized sets of difference
sequences, their duals and matrix transformations (unpublished).
[9] MADDOX, I.J., Continuous and K6the-Teplitz duals of certain sequence spaces, Proc. Camb. Phil.
Soc. 65 (1967), 431-435.
[10] PEYERIMHOFF, A., Ober ein Lemma yon Hem Chow, J. London Math. Soc. 32 (1957), 33-36.