DIFFERENCE SEQUENCE SPACES

In [1] 
Sr(Δ):={x=(xk):(kr|Δxk|)k=1∞∈c0} 
 
for r≥1 is studied. In this paper, we generalize this space to Sr(p,Δ) 
 for a sequence of strictly positive 
reals. We give a characterization of the matrix classes 
(Sr(p,Δ),l∞) and (Sr(p,Δ),l1).


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 Z -1" Y {z 6 u" zz (akz)F 6 Y}.
For any subset X ofw, the sets X= N (z-e) zEX are called the a-and/-duals of X.In this paper we extend the space ,5, (A) to ,5 (p, A) in the same manner as co, c, oo were extended to 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 whe co(p)'= {x e v-limk...,oo [x[ =0}.
3. MATRIX TRANSFORMATIONS For any infinite complex matrix A write A.
for sequence in the nth ak ),.k=1, we ak k= the row of A. Let X and Y be two subsets of w.By (X, Y), we denote the class of all matrices A such that the series A,(z)= a, kzk converges for all x X and each n N, and the sequence k=l Az (A,(x)),= Y for all x X. THEOREM 3.1.Let p= (pk) be a strictly positive sequnce and r _> 1.Then A (s,0, A),eoo) tfand only if [k=l linear operators A, A -] v --, v by Ax (Axe)?' (x x+)', A-lz A-12:k Z3 such that Let A-l:rl O.