A note on Nakano generalized difference sequence space

Abstract

In this paper, we investigate the necessary conditions on any s-type sequence space to form an operator ideal. As a result, we show that the s-type Nakano generalized difference sequence space X fails to generate an operator ideal. We investigate the sufficient conditions on X to be premodular Banach special space of sequences and the constructed prequasi-operator ideal becomes a small, simple, and closed Banach space and has eigenvalues identical with its s-numbers. Finally, we introduce necessary and sufficient conditions on X explaining some topological and geometrical structures of the multiplication operator defined on X.

Kannan Contraction Operator on the Domain of r . -Cesàro Matrix in ℓ<

“…Theorem 23 [30]. Assume s-type V υ ≔ f f = ðs r ðXÞÞ ∈ R N : X ∈ BðP , QÞ and υð f Þ<∞g: If B s V υ is an operator ideal, then the next setups are confirmed:…”

Orlicz Generalized Difference Sequence Space and the Linked Pre-Quasi Operator Ideal

“…When ψ(w) � w r , then ℓ ψ (Δ m n+1 ) � ℓ r (Δ m n+1 ) studied in [5]. Mohiuddine et al [9] investigated the applications of fractional-order difference operators by constructing Orlicz almost null and almost convergent sequence spaces.…”

“…Definition 5 (see [5]). A subspace of the (sss) Y τ is called a premodular (sss) if there is a function τ: Y ⟶ [ 0, ∞ ) verifying the following conditions:…”

“…By B, we will denote the class of all bounded linear operators between any pair of Banach spaces. Definition 6 (see [5]). A class R⊆B is called an operator ideal if every R(W, Z) � R ∩ B(W, Z) satisfies the following conditions:…”

“…Definition 1 (see [5]). e backward generalized difference Δ m n+1 is called an absolute nondecreasing; if…”