Infinite matrices and invariant means

Abstract:

Let $\sigma$ be a one-to-one mapping of the set of positive integers into itself such that ${\sigma ^p}(n) \ne n$ for all positive integers n and p, where ${\sigma ^p}(n) = \sigma ({\sigma ^{p - 1}}(n)),p = 1,2, \cdots$. A continuous linear functional $\varphi$ on the space of real bounded sequences is an invariant mean if $\varphi (x) \geqq 0$ when the sequence $x = \{ {x_n}\}$ has ${x_n} \geqq 0$ for all n, $\varphi (\{ 1,1,1, \cdots \} ) = + 1$, and $\varphi (\{ {x_{\sigma (n)}}\} ) = \varphi (x)$ for all bounded sequences x. Let ${V_\sigma }$ be the set of bounded sequences all of whose invariant means are equal. If $A = ({a_{nk}})$ is a real infinite matrix, then A is said to be (1) $\sigma$-conservative if $Ax = \{ {\Sigma _k}{a_{nk}}{x_k}\} \in {V_\sigma }$ for all convergent sequences x, (2) $\sigma$-regular if $Ax \in {V_\sigma }$ and $\varphi (Ax) = \lim x$ for all convergent sequences x and all invariant means $\varphi$, and (3) $\sigma$-coercive if $Ax \in {V_\sigma }$ for all bounded sequences x. Necessary and sufficient conditions are obtained to characterize these classes of matrices.