The main object of the paper is to study the inclusions [A
σ, p]⊂[Aσ, q], [A
σ, p]⊂[B
σ, q] and [A
σ, p]∩m⊂[B
σ, q] and to characterizae the spaces M([A
σ, p], [A
σ, q]) and M
0([A
σ, p]
0, [A
σ, q]
0). Here [A
σ, p] and [A
σ, q]
0 are certain spaces of strongly summable sequences in terms of an injection σ: N→N and anonnegative infinite matrix A, and M([A
σ, p], [A
σ, q]) is the space of summability factors between [A
σ, p] and [A
σ, q].
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