$R$-type summability methods, Cauchy criteria, $P$-sets and statistical convergence

Abstract:

A summability method $S$ is called an $R$-type summability method if $S$ is regular and $xy$ is strongly $S$-summable to 0 whenever $x$ is strongly $S$-summable to 0 and $y$ is a bounded sequence. Associated with each $R$-type summability method $S$ are the following two methods: convergence in $\mu$-density and $\mu$-statistical convergence where $\mu$ is a measure generated by $S$. In this note we extend the notion of statistically Cauchy to $\mu$-Cauchy and show that a sequence is $\mu$-Cauchy if and only if it is $\mu$-statistically convergent. Let $W\left ( A \right ) = {\overline A ^{\beta \mathbb {N}}} \cap \beta \mathbb {N}\backslash \mathbb {N}$ for $A \subset \mathbb {N}$ and $\mathcal {K}{\text { = }} \cap \left \{ {W\left ( A \right ):A \subseteq \mathbb {N}{\text {,}}{\chi _A}\;{\text {is}}\;{\text {strongly}}\;S - {\text {summable}}\;{\text {to}}\;1} \right \}$. Then $\mu$-Cauchy is equivalent to convergence in $\mu$-density if and only if every ${G_\delta }$ that contains $\mathcal {K}$ in $\beta \mathbb {N}\backslash \mathbb {N}$ is a neighborhood of $\mathcal {K}$ in $\beta \mathbb {N}\backslash \mathbb {N}$. As an application, we show that the bounded strong summability field of a nonnegative regular matrix admits a Cauchy criterion.