Abstract
In this paper, we obtain the Köthe–Toeplitz duals of the domain of an arbitrary invertible summability matrix E in the space ell_{p}. As a consequence, we apply our results to the Fibonacci and Euler sequence spaces and show that some recent works by Altay, Başar, and Mursaleen (Inf. Sci. 176:1450–1462, 2006) are all the special cases of our results.
Highlights
Introduction and preliminariesLet ω, ∞, p, and c be the sets of all sequences, bounded sequences, p-absolutely summable sequences, and convergent sequences, respectively
One can check that if the map E : Ep → p is onto, the space Ep is linearly isomorphic to p and in such a case the columns of the matrix E–1 form a Schauder basis for Ep, where 1 ≤ p < ∞
3 Special cases In the following we present several special cases of Theorems 2.1–2.3
Summary
In [9], the author defined and studied the domain of an arbitrary invertible summability matrix E = (En,k)n,k≥0 in the space p, i.e., Ep := ( p)E. One can check that if the map E : Ep → p is onto, the space Ep is linearly isomorphic to p and in such a case the columns of the matrix E–1 form a Schauder basis for Ep, where 1 ≤ p < ∞. For the infinite summability matrix E, there may be left or right inverses, or even if both exist, they may not be unique. In this paper we deal with the case in which the left and right inverses are equal, and we denote it by E–1.
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