Let $X$ be a linear space over a field $\mathcal F$ of scalars and let $X_{\omega}$ be the set of all infinite sequences $x = (x_0, x_1, \dots )$, where $x_j\in X$. Let  $A = (A_0,A_1, \dots)$ be a sequence in $L_0(X)$. Consider the weighted difference operator in $X_{\omega}: D_Ax = (x_{n+1}- A_nx_n)$. The scalar cases of weighted difference operators have been investigated, among others, by Przeworska-Rolewicz [3] and Kalfat [6]. In this paper we describe the set of all right inverses and the set of all initial operators for $D_A$. Properties of fundamental right inverses and fundamental initial operators are studied. In particular, we give conditions for a fundamental right inverse to be Volterra and apply this result to solve the corresponding initial value problem.