Multidimensional Matrix Characterization of Asymptotic I2−Equivalent and Ideal for Double Sequences. (26th April 2019) - Savaş, Rabia; Patterson, Richard F.
Abstract: The goal of this article is to studyI 2 − equivalence double nonnegative sequences. To accomplish this we present a series of theorems that are similar to the following: If A is a nonnegative four dimensional summability matrix that maps bounded double sequences toℓ 2 and letI 2 andJ 2 be admissible ideals inN × N then the following are equivalent. Ifx = ( x k, l ) andy = ( y k, l ) are bounded double sequences such thatx ∼ I 2 y, ( x k, l ) ∈ P 0 ″ and( y k, l ) ∈ P δ ″, for someδ > 0, thenR m, n ( A x ) ∼ J 2 R m, n ( A y ) for each m and n, J 2 − lim m, n ∑ k, l ≥ m, n ∑ p, q ∈ S 2 a k, l, p, q ∑ k, l ≥ m, n ∑ ∞, ∞ p, q = 1, 1 a k, l, p, q = 0 for eachS 2 ∈ I 2 . In addition other variations and implications shall also be presented.