Variation of an Element in the Matrix of the First Difference Operator and Matrix Transformations

We are interested in the study of the first difference operator. This one can be represented by the infinite matrix ∆. Many authors have given results on this last operator, see for instance Malkowsky [8],[9], Kizmaz [2], Çolak and Et [1] and more recently de Malafosse [6]. These authors gave many characterizations of the operators A mapping the space (∆μ)−1 (l∞) into l∞, that is A ∈ (l∞ (∆) , l∞). Malkowsky [8], [9] and Malkowsky and Parashar [7] found new Schauder bases in the spaces c0 (∆) and c (∆). They gave many results concerning AK and BK spaces considering the Λ-strongly null and Λstrongly convergent sequences and have studied extensions of some results given by Wilansky [12]. Note that many authors have dealt with the Cesàro operator and there is a simple relation between this operator and the operator represented by ∆. Recall that the spectrum of the Cesàro operator C1 in certain spaces has been studied by Reade [11], Okutoyi [10] and de Malafosse [5]. Here are recalled some properties of ∆ considered as an operator from the space sα into itself. Further, as in [5], we deal with matrix perturbation and consider the new matrix ∆pq (a ′) obtained from ∆ by changing only one element in the p-th row and in the q-th column of the infinite matrix and deduce some results on the spaces sα (( ∆pq (a′) )μ). Then we deal with matrix transformations between matrix domains such as sα (( ∆pq (a ′) )μ) or sβ . The paper is organized as follows. In the second section we recall some results and definitions concerning the infinite matrix theory. In the third section some properties of the spaces sr (∆), sr ( (∆+) ) and s1 (∆) are given. Next, we