We can describe the norm for an operator given as T : X → Y as follows: It is the most appropriate value of U that satisfies the following inequality ∥Tx∥ ≤ U∥x∥ Y X and also for the lower bound of T we can say that the value of L agrees with the following inequality ∥Tx∥ ≥ L∥x∥ , Y X where ∥.∥ and ∥.∥ stand for the norms corresponding to the spaces X and Y. The main feature of this X Y article is that it converts the norms and lower bounds of those matrix operators used as weighted sequence space ℓ (w) into a new space. This new sequence space is the generalized weighted sequence space. For p this purpose, the double sequential band matrix B˜ (r˜, s˜) and also the space consisting of those sequences whose B˜ (r˜, s˜) transforms lie inside ℓ (w˜ ), where r˜ = (r ), s˜ = (s ) are convergent sequences of positive real p n n numbers. When comparing with the corresponding results in the literature, it can be seen that the results of the present study are more general and comprehensive.