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Matrix summability of statistically convergent sequences. (English) Zbl 0801.40005
Let $K$ be an index set and $\varphi\sp K:= (\varphi\sb j\sp K)$ be defined by $\varphi\sb j\sp K=1$ $(j\in k)$, $\varphi\sb j\sp K=0$ otherwise. For a non-negative regular $(c\to c)$ matrix $A$, the $A$- density of $K$ is $\delta\sb A(K):= \lim\sb A \varphi\sp K$ whenever $\varphi\sp K\in c\sb A$. For a sequence $x=(x\sb k)\sb{k\geq 1}$ and a number $x\sb 0$, denote $K\sb \varepsilon:= \{k$: $\vert x\sb k- x\sb 0\vert \geq \varepsilon\}$; then $x$ is said to be $A$-statistically convergent to $x\sb 0$ $(x\in \text{st}\sb A)$ if $\delta\sb A (K\sb \varepsilon)=0$ for every $\varepsilon>0$. We also denote the $K$-section of $x$ as $x\sp{[K]}$, where $x\sb k\sp{[K]}= x\sb k$ $(k\in K)$, $x\sb k\sp{[K]}=0$ otherwise, and a sequence space $X$ is called section-closed if $x\sp{[K]} \in x$ for all $x\in X$ and for every index set $K$. The $K$-column-section of a matrix $B$ is denoted by $B\sp{[K]}:= (b\sb{nk}\sp{[K]})$, where $b\sb{nk}\sp{[K]}= b\sb{nk}$ $(k\in K)$, $b\sb{nk}\sp{[K]} =0$ otherwise. For any two sequence spaces $W$, $Y$, we denote by $(W,Y)$ the set of all matrices $H$ which map $W$ into $Y$ $(W\subseteq Y\sb H)$. The main Theorem 4.1 shows: Let $X$ be a section- closed sequence space containing $e:= (1,1,\dots)$ and let $Y$ be an arbitrary sequence space; then $B\in (X\cap \text{st}\sb A,Y)$ if and only if $B\in (X\cap c, Y)$ and $B\sp{[K]}\in (X,Y)$ $(\delta\sb A (K) =0)$. There are some modifications and corollaries, and a final section which considers matrix maps of statistically convergent bounded sequences (where we take $X=\ell\sb \infty$).

40C05Matrix methods in summability
40D25Inclusion theorems; equivalence theorems
40J05Summability in abstract structures
46A45Sequence spaces
40A05Convergence and divergence of series and sequences