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