Some multiplier difference sequence spaces defined by a sequence ...

184 Hemen Dutta(x k ) belongs to Z(F, ∆ 1 (3) , Λ, p), but (y k) does not belong to Z(F, ∆ 1 (3), Λ, p)for Z = l ∞ , c and c 0 . Hence the spaces Z(F, ∆ n (m) , Λ, p) for Z = l ∞, c andc 0 are not convergence free in general.Theorem 8 The spaces c 0 (F, ∆ n (m) , Λ, p), c(F, ∆n (m) , Λ, p) and l ∞(F, ∆ n (m) ,Λ, p) are not sequence algebra in general.Proof. The proof follows from the following examples.Example 7 Let n = 2, m = 1, p k = 1 for all k and f k (x) = x 22 , for eachk ∈ N and x ∈ [0, ∞). Then ∆ 2 (1) λ kx k =λ k x k −2λ k−1 x k−1 +λ k−2 x k−2 , for allk ∈ N. Consider Λ = ( 1k 4 ) and let x = (k 5 ) and y = (k 6 ). Then x, y belongto Z(F, ∆ 2 (1) , Λ, p) for Z = l ∞, c, but x.y does not belong to Z(F, ∆ 2 (1), Λ, p)for Z = l ∞ , c. Hence the spaces c(F, ∆ n (m) , Λ, p), l ∞(F, ∆ n (m), Λ, p) are notsequence algebra in general.Example 8 Let n = 2, m = 1, p k = 3 for all k and f k (x) = x 5 , for eachk ∈ N and x ∈ [0, ∞). Then ∆ 2 (1) λ kx k =λ k x k −2λ k−1 x k−1 +λ k−2 x k−2 , for allk ∈ N. Consider Λ = ( 1k 7 ) and let x = (k 8 ) and y = (k 8 ). Then x, y belongto c 0 (F, ∆ 2 (1) , Λ, p), but x.y does not belong to c 0(F, ∆ 2 (1) , Λ, p) for Z = l ∞, c.Hence the space c 0 (F, ∆ n (m), Λ, p) is not sequence algebra in general.References[1] M. Et and R. Colak, On generalized difference sequence spaces, SoochowJ. Math. 21, 1985, 147-169.