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Let C0 denote the set of all
complex null sequences, and let S0 denote the set of all sequences in C0 which have
at most a finite number of zero terms. If a = {ap}∈ S0 and b = {bp}∈ S0, we say
that a converges faster than b,a < b, provided limap∕bp = 0. We say that a
and b converge at the same rate, a ∼ b, provided 0 < lim liminf |ap∕bp| and
limsup|ap∕bp| < ∞. If a ∈ S0, let [a] = {x ∈ S0 : x ∼ a}. Let E0 = {[x] : x ∈ S0}. If
[a],[b] ∈ E0, then we say that [a] is less than [b],[a] < ′[b], provided a < b. We note
that E0 is partially ordered with respect to ≦′. In this paper we study matrix
summability over subsets of S0 and over elements of E0. Open intervals in S0 will be
denoted by (a,b),(a,−), and (−,b), where (a,−) = {iχj ∈ S0 : a < x} and
(−,b) = {x ∈ S0 : x < b}. Some of our results characterize, for matrices, maximal
summability intervals in S0. Such intervals are of the form (−,b), never of the form
(−,b] = {x ∈ S0 : either x < b or x ∼ b}.
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