Summability field of a matrix method

Remarks

The set $c_A$ is a sequence space, since both conditions are preserved by sums and scalar multiples.

Example: for the identity matrix, $(Ax)_n = x_n$, so $c_A$ is the space $c$ of convergent sequences.

Example: for the Cesàro matrix, $a_{nk} = 1/(n+1)$ for $k \le n$ and $a_{nk} = 0$ for $k > n$, the sequence $x = (1, 0, 1, 0, \dots)$ lies in $c_A$. Indeed $(Ax)_n$ is the proportion of ones among $x_0, \dots, x_n$, which tends to $1/2$.

Non-example: for the same matrix, $x_k = (-1)^k k$ is not in $c_A$. The sum $x_0 + \dots + x_n$ equals $n/2$ for even $n$ and $-(n+1)/2$ for odd $n$, so $(Ax)_n$ comes close to $1/2$ and to $-1/2$ infinitely often.

A matrix $A$ is regular when $c \subseteq c_A$ and $\lim_n (Ax)_n = \lim_k x_k$ for every $x \in c$. With a suitable topology, $c_A$ is an FK-space.