Asymptotic equivalence and summability

This paper is a study of the relatioship. between the asymptotic equivalence of two sequences (limnxn/yn=1) and three variatious of this equivalence for a sequence-to-sequence transformation A, the three variations are give by the ratios RmAx/RmAy. SmAx/SmAy, And mAx/ mAy, where RmAz:= ￠ ‘n ￠ ‰ ￥m|(Az)n|, SmAz:= ￠ ‘n ￠ ‰ ¤m|(Az)n|, and mAz:=supn ￠ ‰ ￥m|(Az)n|.