On the Domain of the Four-Dimensional Sequential Band Matrix in Some Double Sequence Spaces

Orhan Tuğ,Vladimir Rakočević,Eberhard Malkowsky

doi:10.3390/math8050789

Abstract

Let E represent any of the spaces M u , C ϑ ( ϑ = { b , b p , r } ) , and L q ( 0 < q < ∞ ) of bounded, ϑ -convergent, and q-absolutely summable double sequences, respectively, and E ˜ be the domain of the four-dimensional (4D) infinite sequential band matrix B ( r ˜ , s ˜ , t ˜ , u ˜ ) in the double sequence space E, where r ˜ = ( r m ) m = 0 ∞ , s ˜ = ( s m ) m = 0 ∞ , t ˜ = ( t n ) n = 0 ∞ , and u ˜ = ( u n ) n = 0 ∞ are given sequences of real numbers in the set c ∖ c 0 . In this paper, we investigate the double sequence spaces E ˜ . First, we determine some topological properties and prove several inclusion relations under some strict conditions. Then, we examine α -, β ( ϑ ) -, and γ -duals of E ˜ . Finally, we characterize some new classes of 4D matrix mappings related to our new double sequence spaces and conclude the paper with some significant consequences.

Highlights

The set of all complex valued double sequences is expressed as Ω := {x = : xmn ∈ C, ∀m, n ∈N}, which is a vector space that possesses scalar multiplication and coordinate-wise addition, where C is the complex field, and N = {0, 1, 2, ...}
We investigate the double sequence spaces E
Mu, C p, Cbp, Cr, and Lq denote the classical spaces of all double sequences that are bounded, convergent in the Pringsheim sense, convergent in the Pringsheim sense and bounded, regular convergent, and q-absolutely summable, respectively, where 0 < q < ∞

Summary

Introduction

The set E A , which is the θ-summability domain of a 4D infinite matrix A in a double sequence space E, is defined by: EA =. Studied the topological properties, inclusion relations, dual spaces, and matrix transformations of some new spaces of double sequences. We introduce some new double e(Mu ), B e(C p ), B e(Cbp ), B e(Cr ), and B e(Lq ),(0 < q < ∞) whose 4D sequential band sequence spaces B e matrix B-transforms are in the spaces Mu , C p , Cbp , Cr , andLq , respectively. The goal of the following sections of this paper is to provide extensive proofs of some topological properties, inclusion relations, dual spaces of these new double sequence spaces, and the characterization of some 4D infinite matrix classes.

New Double Sequence Spaces e in the spaces E are introduced and denoted by

Matrix Transformations Related to the Sequence Spaces B and B