## Double sequence core theorems.(English)Zbl 0949.40007

Let $$s=\{s_n\}$$ denote a complex sequence. Define $$R_n$$ to be the smallest closed convex region of the complex plane which contains $$\{s_n, s_{n+1}, \dots\}$$, $$R:=\bigcap^\infty_{n=1} R_n$$. The set $$R$$ is called the core of $$s$$. Knopp’s core theorem states that, if $$A$$ is a nonnegative regular matrix, then the core of the $$A$$-transform of $$s$$ is contained in the core of $$s$$. For double sequences, the usual convergence used is that of Pringsheim, which leads to the Toeplitz conditions of regularity of a four-dimensional matrix (written RH-regular), for convergent sequences which are also bounded. In this paper the author defines the concept of a core sequence for double sequences and proves the two-dimensional analog of Knopp’s core theorem. The notation $$P$$-limit means the limit in the Pringsheim sense. The author also proves that, for all real-valued double sequences $$[x]$$, the condition $$P$$-$$\lim \sup[Ax] \leq P$$-$$\lim\sup [x]$$ is equivalent to $$A$$ being an RH-regular matrix satisfying $$P$$-$$\lim_{m,n} \sum^\infty_{k,j=0} |a_{mnjk} |=1$$.

### MSC:

 40B05 Multiple sequences and series 40C05 Matrix methods for summability
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