Some classes of matrix transforms related to speed-Maddox spaces over ultrametric fields

Abstract

Let K be a complete, non-trivially valued, ultrametric (or non-archimedean) field, and λ = {λ_n} − a sequence in K with the property 0 < |λ_n| ↗ ∞, n ➝ ∞, i.e., the speed of convergence. In the present paper, the concepts of speed-Maddox space, paranormed zero-convergence, paranormed convergence and paranormed boundedness with speed λ (or shortly, paranormed λ-zero-convergence, paranormed λ-convergence and paranormed λ-boundedness) over K have been recalled. Let μ be another speed in K. Necessary and sufficient conditions are found for a matrix A over K to transform the set of all paranormally λ-bounded sequences into the set of all paranormally μ-bounded, all paranormally μ-convergent or all paranormally μ-zero-convergent sequences.