Calculations on some sequence spaces

where B = (bn)n≥1 is a one-column matrix and X the unknown, see [1, 2, 3, 4, 5, 6, 7, 8, 10]. Equation (1.2) can be written in the form AX = B, where AX = (An(X))n≥1. In this paper, we will also consider A an operator from a sequence space into another sequence space. A Banach space E of complex sequences with the norm ‖‖E is a BK space if each projection PnX = xn is continuous for all X ∈ E. A BK space E is said to have AK, (see [12, 13]), if B =∑∞m=1bmem, for every B = (bn)n≥1 ∈ E, (with en = (0, . . . ,1, . . .), 1 being in the nth position), that is, ∥∥∥∥ ∞ ∑