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$f$-conservative matrix sequences. (English) Zbl 0748.40002
Let ${\cal A}$ denote the sequence of real matrices $A\sb p=(a\sb{nk}(p))$. For a sequence $x=(x\sb k)$, $(Ax)\sp p\sb n=\sum\sb ka\sb{nk}(p)x\sb k$ if it exists for each $n,p$ and $Ax=((Ax)\sp p\sb n)\sp \infty\sb{n,p=0}$. A sequence $x$ is said to be $A$-summable to $x\sb 0$ if $\lim\sb n(Ax)\sp p\sb n=x\sb 0$ uniformly in $p$. This encompasses the usual summability method $A$, ordinary convergence and among other methods almost convergence as introduce by {\it G. G. Lorentz} [Acta Math. Uppsala 80, 167-190 (1948; Zbl 0031.29501)] Let $A=(a\sb{n,k})$ be an infinite matrix of real numbers $a\sb{n,k}(n,k=0,1,\ldots)$ and $\lambda,\mu$ be two non-empty subsets of the space $s$ of all real sequences. The matrix $A$ defines a transformation from $\lambda$ into $\mu$, if for every sequence $x=(x\sb k)\in\lambda$ the sequence $Ax=((Ax)\sb n)$ exists and is in $\mu$ where $(Ax)\sb n=\sum\sb ka\sb{n,k}x\sb k$. By $(\lambda:\mu)$ all such matrices are denoted. Let $f,fs$ denote the spaces of all almost convergent real sequences and series respectively. The main purpose of this paper is to determine the necessary and sufficient conditions on the matrix sequence $A=(A\sb p)$ in order that $A$ is contained in one of the classes $(f:f)$, $(f:fs)$, $(fs:f)$ and $(fs:fs)$.

40C05Matrix methods in summability