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January 18, 2026Publications de l Institut Mathematique0 citationsOpen Access

A Steinhaus-type theorem for multi-dimensional matrix transformations in (L₁, L₁)

Key Points

The aim is to prove a Steinhaus-type theorem for multidimensional matrix transformations, particularly in (L_1, L_1).
Proved a theorem for four-dimensional matrix transformations of double sequences.
Investigated properties of various sequence spaces such as bounded variation and convergence.
Provided a counterexample illustrating the failure of the theorem in non-archimedean fields.
Established (L_1, L_1, P) ∩ (L_s, L_1) = ∅ for s > 1.
The theorem extends classical results on single sequences to four dimensions.
Demonstrated the limitations of the theorem with a counterexample.

Abstract

We prove a Steinhaus-type theorem for four-dimensional matrix transformations of double sequences, establishing that (L₁, L₁, P) (Lₛ, L₁) =, s > 1. This extends Fridy? s classical result for single sequences. Our results hold for sequences of bounded variation, bounded, Pringsheim convergent, bounded Pringsheim convergent, and regularly convergent sequence spaces. We show this theorem fails in non-archimedean fields through a counterexample.

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