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

Some multiplier difference sequence spaces 183Theorem 6 The spaces c 0 (F, ∆ n (m) , Λ, p), c(F, ∆n (m) , Λ, p) and l ∞(F, ∆ n (m) ,Λ, p) are not symmetric in general.Proof. The proof follows from the following example.Example 5 Let n = 2, m = 2, p k = 2 for all k odd and p k = 3 for allk even and f k (x) = x 2 , for all x ∈ [0, ∞) and for all k ≥ 1. Then∆ 2 (2) λ kx k =λ k x k − 2λ k−2 x k−2 + λ k−4 x k−4 , for all k ∈ N. Consider thesequences Λ = (1, 1, 1, . . . ) and x = (x k ) defined as x k = k for k oddand x k = 0 for k even. Then ∆ 2 (2) λ kx k = 0, for all k ∈ N. Hence(x k ) ∈ Z(F, ∆ 2 (2) , Λ, p) for Z = l ∞, c and c 0 . Consider the rearranged sequence,(y k ) of (x k ) defined as(y k ) = (x 1 , x 3 , x 2 , x 4 , x 5 , x 7 , x 6 , x 8 , x 9 , x 11 , x 10 , x 12 , ...)Then (y k ) does not belong to Z(F, ∆ 2 (2) , Λ, p) for Z = l ∞, c and c 0 .Hence the spaces Z(F, ∆ n (m) , Λ, p) for Z = l ∞, c and c 0 are not symmetricin general.Theorem 7 The spaces c 0 (F, ∆ n (m) , Λ, p), c(F, ∆n (m) , Λ, p) and l ∞(F, ∆ n (m) ,Λ, p) are not convergence free in general.Proof. The proof follows from the following example.Example 6 Let m = 3, n = 1, p k = 6 for all k and f k (x) = x 3 , for keven and f k (x) = x, for k odd, for all x ∈ [0, ∞). Then ∆ 1 (3) λ kx k =λ k x k −λ k−3 x k−3 , for all k ∈ N. Let Λ = ( 7) and consider the sequences (x k k) and(y k ) defined as x k = 4k for all k ∈ N and y 7 k = 1 7 k3 for all k ∈ N. Then