Abstract
We investigate some new topological properties of the multiplication operator on C(p) defined by Lim (Tamkang J. Math. 8(2):213–220, 1977) equipped with the pre-quasi-norm and the pre-quasi-operator ideal formed by this sequence space and s-numbers.
Highlights
1 Introduction Throughout the article, we denote the space of all bounded linear operators from a Banach space X into a Banach space Y by L(X, Y ) and if X = Y, we write L(X), the space of all complex sequences by w, the real numbers R, the complex numbers C, N = {0, 1, 2, . . .}, the space of convergent complex sequences to zero by C0, the space of bounded complex sequences by ∞ and all sequences whose its elements are complex by CN
Definition 4.1 Let β ∈ CN be a bounded sequence and E be a pre-quasi-normed, the multiplication operator is defined as Tβ : E → E, where Tβ x = βx =∞ k=0, for all x ∈ E
This proves {edn : dn ∈ Bδ} is a bounded sequence, which cannot have a convergent subsequence under Tβ. This shows that Tβ cannot be a compact, is not approximable operator, which is a contradiction
Summary
Throughout the article, we denote the space of all bounded linear operators from a Banach space X into a Banach space Y by L(X, Y ) and if X = Y , we write L(X), the space of all complex sequences by w, the real numbers R, the complex numbers C, N = {0, 1, 2, . . .}, the space of convergent complex sequences to zero by C0, the space of bounded complex sequences by ∞ and all sequences whose its elements are complex by CN. For a sequence (pn) with infn pn > 0, Lim (see [1]) defined and studied the sequence space C(p) as follows: C(p) = x = (xn) ∈ ω : ρ(βx) < ∞ for some β > 0 , ≤1 . C(p) = (xi) ∈ ω :
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