On some second order Cesaro difference spaces of non-absolute type

The second order Cesaro sequence spaces of non-absolute type Xp(¢2) for 1 • p • 1 were defined and studied in (1). It seems, however, that the characterizations of their fl-duals given there do not hold for 1 < p • 1. In this paper, we determine the fl-duals (Xp(¢2))fl for 1 • p • 1. k = 0 (k 6= n). We write n ‚ = (n‚)1 n=1, k‚ = (k‚)1 k=1 for ‚ 2 IR and 1=n = (1=n)1 n=1. Let x;y 2 ! and X ‰ !. We write xy = (xkyk)1 k=1, x ¡1 ⁄ Y = fa 2 ! : ax 2 Y g, xfl = x¡1 ⁄ cs and Xfl = \ x2X xfl = fa 2 ! : P1 k=1akxk converges for all x 2 Xg for the fl-dual of X. Given any infinite matrix A = (ank)1 n;k=1 of complex numbers and any sequence x, we write An = (ank)1 k=1 for the sequence in the n-th row of A, An(x) = P1 k=1ankxk (n = 1;2;:::) and A(x) = (An(x)) 1 n=1, provided An 2 xfl for all n. Furthermore, XA = fx 2 ! : A(x) 2 Xg denotes the matrix domain of A in X. We define the matrices §, ¢ and E by §nk = 1 (1 • kn), §nk = 0 (k > n), ¢nn = 1, ¢n;n+1 = ¡1, ¢nk = 0 (otherwise), enk = 1 (k ‚ n + 1) and enk = 0 (1 • kn) for all n, and write ¢2 = ¢¢. Then the sets Xp(¢2) = ((1=n)¡1 ⁄ 'p)§¢2 are the second order Cesaro