B-statistical core and ideal core of double sequences in 2-normed spaces via RH-regular families

OpenAlex İngilizce

We investigated geometric limit sets of double sequences in finite dimensional $ 2 $-normed spaces when negligibility of index sets was measured by a density induced by a nonnegative Robison–Hamilton (RH)-regular family $ \mathfrak{B} $ of four-dimensional matrices. First, using the $ \mathfrak{B} $-density, we defined $ \mathfrak{B} $-statistical limit superior and limit inferior for the scalar reductions generated by the seminorms $ x\mapsto\|x, u\| $, and we studied the associated real cluster behavior. Next, we introduced $ \mathfrak{B} $-statistical cluster points and the $ \mathfrak{B} $-statistical core of a double sequence as the intersection of all closed convex sets that contained the sequence outside a $ \mathfrak{B} $-density zero set. We obtained a disk-type representation of the core via intersections of sets of the form $ \{x\in X:\|x-z, u\|\le r\} $, where the radii were governed by $ \mathfrak{B} $-statistical $ \limsup $. We also proved a Knopp-type inclusion theorem: for a family of transforms satisfying a natural $ \mathfrak{B} $-regularity condition, the Knopp core of each transform was contained in the $ \mathfrak{B} $-statistical core of the original sequence. Finally, replacing $ \mathfrak{B} $-density zero sets by a strongly admissible ideal $ \mathcal{I}_2 $ on $ \mathbb{N}\times \mathbb{N} $, we defined ideal cores, established disk representations, compared the density-based and ideal cores, and identified an explicit condition under which the $ \mathcal{I}_2 $-core and the $ \mathcal{I}_2^\ast $-core coincided.