Domain of the double sequential band matrix in the classical sequence spaces

Abstract

Let λ denote any one of the classical spaces ∞ , c, c 0 and p of bounded, convergent, null and absolutely p-summable sequences, respectively, and λ also be the domain of the double sequential band matrix B( r, s) in the sequence space λ, where (r n ) ∞ n=0 and (s n ) ∞ n=0 are given convergent sequences of positive real numbers and 1 ≤ p < ∞. The present paper is devoted to studying the sequence space λ. Furthermore, the β-and γ -duals of the space λ are determined, the Schauder bases for the spaces c, c 0 and p

Euler-Ces`aro difference spaces of bounded, convergent and null sequences

“…∆ c 0 , c and ℓ ∞ c 0 (∆), c(∆) and ℓ ∞ (∆) [16] R t c 0 , c and ℓ ∞ r t 0 , r t c and r t ∞ [17,28] B (r, s) c 0 , c and ℓ ∞ c 0 , c and ℓ ∞ [18] B (r, s, t ) c 0 , c and ℓ ∞ c 0 (B ), c(B ) and ℓ ∞ (B ) [19] C 1 c 0 , c and ℓ ∞ c 0 , c and X ∞ [27,26] A r c 0 and c a r 0 and a r c [29] E r c 0 , c and ℓ ∞ e r 0 , e r c and e r ∞ [31,30] ∆ 2 c 0 and c c 0 (∆ 2 ) and c(∆ 2 ) [32] u∆ 2 c 0 and c c 0 (u; ∆ 2 ) and c(u; ∆ 2 ) [33] ∆ m c 0 and c c 0 (∆ m ) and c(∆ m ) [34,35] R q c 0 and c (N , q) 0 and (N , q) [36] ∆ (m) c 0 and c c 0 (∆ (m) ) and c(∆ (m) ) [37] G(u, v) c 0 , c and ℓ ∞ c 0 (u, v), c(u, v) and ℓ ∞ (u, v) [38] Λ c 0 and c c λ 0 and c λ [39] B ( r , s) c 0 , c and ℓ ∞ c 0 , c and ℓ ∞ [40] A λ c 0 and c A λ (c 0 ) and A λ (c) [41] F c 0 and c c 0 ( F ) and c( F ) [42] N t c 0 , c and ℓ ∞ c 0 (N t ), c(N t ) and X a(p) [43,44] In 1978, the domain of Cesàro matrix C 1 of order one in the classical sequence spaces ℓ ∞ and ℓ p were introduced by Ng and Lee [26], where 1 ≤ p < ∞. Following Ng and Lee [26], Sengönül and Başar [27] have studied the domain of Cesàro matrix C 1 of order one in the classical sequence spaces c 0 and c. In 1978, the domain of Nörlund matrix N t in the classical sequence spaces ℓ ∞ and ℓ p were introduced by Wang [44], where 1 ≤ p < ∞.…”

On the Domain of the Fibonacci Difference Matrix

“…(13) R = (r nk ) ∈ (bs, bv 0 ) iff Equations (21), (18) and (21) hold [35]. (14) R = (r nk ) ∈ (cs, c 0 ) iff Equations (11) and (12) hold with a k = 0 for all k ∈ N [40]. (15) R = (r nk ) ∈ (cs, bs) iff Equations (9) and (22) hold [38].…”

The method of almost convergence with operator of the form fractional order and applications

“…In the book of Basar [13], it can be seen that the qualified studies about matrix domain(also, [4]- [12], [18], [19], [20], [31], [33]- [36]).…”