Core Theorems for Subsequences of Double Complex Sequences

Abstract

In this article we present core theorems for double sequences whose entries are complex numbers. These results extend work of Miller and Patterson dealing with double sequences of real numbers. The proofs in this paper are much more involved then the proofs in the article just mentioned as the convex sets in the plane are, in general, much more involved then the trivial convex sets in the line. We give an answer to the following question. If $w$ is a bounded double sequence with complex entries and $A$ is a $4$-dimensional matrix summability method, under what conditions on $A$ does there exist $z$, a subsequence (rearrangement), of $w$ such that each complex number $t$, in the core of $w$, is a limit point of $Az$?