Certain topological properties and duals of the domain of a triangle matrix in a sequence space

Abstract

The matrix domain of the particular limitation methods Cesàro, Riesz, difference, summation and Euler were studied by several authors. In the present paper, certain topological properties and β-and γ -duals of the domain of a triangle matrix in a sequence space have been examined as an application of the characterization of the related matrix classes. Preliminaries, background and notationBy a sequence space, we understand a linear subspace of the space w = C N of all complex sequences which contains φ, the set of all finitely non-zero sequences, where N = {0, 1, 2, . . .}. We write ∞ , c and c 0 for the sequence spaces of all bounded, convergent and null sequences, respectively. Also by p , we denote the space of all absolutely p-summable sequences, where 1 p < ∞. bv is the space consisting of all sequences (x k ) such that (x k − x k+1 ) in 1 and bv 0 is the intersection of the spaces bv and c 0 . We shall assume throughout unless stated otherwise that p, q > 1 with p −1 + q −1 = 1 and 0 < r < 1, and use the convention that any term with negative subscript is equal to naught.Let A = (a nk ) be an infinite matrix of complex numbers a nk , where n, k ∈ N, and write

Integrated and Differentiated Sequence Spaces

“…for all n, k ∈ N. Then we deduce from Lemma 3.3 (ii) with (3.19) that ax = (a k x k ) ∈ cs whenever x = (x k ) ∈ ∫ bv if and only if Gy ∈ c whenever y = (y k ) ∈ ℓ 1 . We obtain from Lemma 3.1 and Lemma 3.2, the result that a = (a k ) ∈ ( Theorem 3.7.…”

“…for all k, n ∈ N. It follows from (3.18) with Lemma 3.3 (iii) that ax = (a n x n ) ∈ ℓ 1 whenever x = (x k ) ∈ ∫ bv if and only if Fy ∈ ℓ 1 whenever y ∈ ℓ 1 . This means that a = (a n ) ∈ (…”

Matrix transformation of Fibonacci band matrix on generalized $bv$-space and its dual spaces

“…We refer the reader to [2,3,4,5,11,16] for the concept of matrix domain. Define the sequence {f n } ∞ n=0 of Fibonacci numbers given by the linear recurrence relations f 0 = f 1 = 1 and f n = f n−1 + f n−2 , n ≥ 2.…”

Some properties of Generalized Fibonacci difference bounded and $p$-absolutely convergent sequences

“…The concept of matrix domain we refer to [2,3,4,9,12,13,14,15,16,17,18]. Define the sequence {f n } ∞ n=0 of Fibonacci numbers given by the linear recurrence relations f 0 = f 1 = 1 and f n = f n−1 + f n−2 , n ≥ 2.…”

“…, where cs and bs are the sequence spaces of all convergent and bounded series, respectively (see for instance [2,7,15]). Now let A = (a nk ) be an infinite matrix and consider the following conditions:…”