Ordinary, absolute and strong summability and matrix transformations

Abstract

Abstract. Many important sequence spaces arise in a natural way from various concepts of summability, namely ordinary, absolute and strong summability. In the first two cases they may be considered as the domains of the matrices that define the respective methods of summability; the situation, however, is different and more complicated in the case of strong summability.Given sequence spaces X and Y , we find necessary and sufficient conditions for the entries of a matrix to map X into Y , and characterize the subclass of those matrices that are compact operators.This paper gives a survey of recent research in the field of matrix transformations at the University of Niš, Serbia and Montenegro, in the past four years.

Matrix transformations and generators of analytic semigroups

“…A Banach space X ⊂ s is a BK space if the projection P n : x → x n from X into C is continuous for all n. A BK space X ⊃ ϕ is said to have AK if for every x ∈ X, x = lim p→∞ p k=1 x k e k , where e k = (0,...,1,...), 1 being in the kth position. It is well known that if X has AK, then Ꮾ(X) = (X,X), see [9,13,19].…”

On the Banach algebra ℬ(lp(α))

“…A BK space E is said to have AK if every sequence X = (x n ) ∞ n=1 ∈ E has a unique representation X = ∞ n=1 x n e n , where e n is the sequence with 1 in the nth position and 0 otherwise. It is well known [9] that if E has AK, then Ꮾ(E) = (E, E).…”

Difference sequence spaces derived by using generalized means

“…Since A(r, s, t) is a triangle, it has a unique inverse and the inverse is also a triangle [7]. Take…”