---
tag: C22331MBY7
kind: definition
name: Summability field of a matrix method
msc: [40C05, 40H05]
defines: 799DX4375M
uses: [M6311DWJ1E, XYTN19EPF2, 0W4CN9E2W5, 29C7C41A52, 71DMQHW6JZ]
sources:
  - url: https://en.wikipedia.org/wiki/Silverman%E2%80%93Toeplitz_theorem
---

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $\omega$ be the space of all sequences in $\mathbb{K}$.
Let $A = (a_{nk})_{n, k \ge 0}$ be an infinite matrix with entries in $\mathbb{K}$, viewed as a [[matrix summability method|M6311DWJ1E]].
For $x = (x_k)_{k \ge 0} \in \omega$, write $(Ax)_n = \sum_{k \ge 0} a_{nk} x_k$ when this series converges.
The *summability field* (or *convergence domain*) of $A$ is
$$c_A = \set{x \in \omega : (Ax)_n \text{ converges for every } n, \text{ and } \lim_{n \to \infty} (Ax)_n \text{ exists}}.$$

## Remarks

The set $c_A$ is a [[sequence space|XYTN19EPF2]], 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|0W4CN9E2W5]].

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|29C7C41A52]] 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|71DMQHW6JZ]].