Four dimensional matrix characterizations of the Pringsheim core
In an earlier work [Int.~J.~Math.~Math.~Sci.~ 22, 785--793 (1999; Zbl 0949.40007)], the author defined the Pringsheim core, \(P-C \{x\}\), of a double sequence \([x]\), to be \(\bigcap^\infty_{n=1} P-C_n \{x\}\), where \(P-C_n\{x\}\) is the least, closed convex set that includes all points \(x_{kl}\), with both \(k\) and \(l\) greater than \(n\), and proved the following analog of the Knopp core theorem:\ If \(A\) is a nonnegative, four-dimensional, RH-regular summability matrix, then \(P-C\{Ax\}\subseteq P-C\{x\}\), for every bounded sequence for which \([Ax]\) exists. In the present article, the author defines absolute equivalency of two RH-regular matrices, for a given class of double sequences, gives necessary and sufficient conditions for RH-regular matrices to be absolutely equivalent, for all bounded sequences, and, finally, establishes the following result: The Pringsheim core of \(A\{s\}\), where \([s]\) is a bounded (double) sequence, is contained in the core of \([s]\) if and only if \(A\) is RH-regular and is absolutely equivalent to a nonnegative matrix \(B\), for all bounded sequences.
scientific article; zbMATH DE number 2119030