Small Pre-Quasi Banach Operator Ideals of Type Orlicz-Cesáro Mean Sequence Spaces

Abstract

In this paper, we give the sufficient conditions on Orlicz-Cesáro mean sequence spaces cesφ, where φ is an Orlicz function such that the class Scesφ of all bounded linear operators between arbitrary Banach spaces with its sequence of s-numbers which belong to cesφ forms an operator ideal. The completeness and denseness of its ideal components are specified and Scesφ constructs a pre-quasi Banach operator ideal. Some inclusion relations between the pre-quasi operator ideals and the inclusion relations for their duals are explained. Moreover, we have presented the sufficient conditions on cesφ such that the pre-quasi Banach operator ideal generated by approximation number is small. The above results coincide with that known for cesp (1<p<∞).

Eigenvalues of s-type operators on $C(p)$ equipped with a pre-quasi norm

“…Definition 2.5 (see [23]) A class of linear sequence spaces X is called a special space of sequences (sss) if…”

“…Definition 2.6 (see [23]) A subclass of the special space of sequences called a pre-modular (sss) if there is a function : X → [0, ∞[ satisfying the following conditions: (i) (u) ≥ 0 for each u ∈ X and (u) = 0 ⇔ u = θ , where θ is the zero element of X, (ii) there exists L ≥ 1 such that (βu) ≤ L|β| (u) for all u ∈ X, and for any scalar β,…”

“…The concept of pre-quasi-operator ideal which is more general than the usual classes of operator ideal. Definition 2.7 (see [23]) A function g : → [0, ∞) is said to be a pre-quasi-norm on the ideal if the following conditions holds:…”

Some Properties of Prequasi Normed Generalized de La Vallée Poussin’s Mean Sequence Space

“…(1) If γ i = i + 1, for every i ∈ ℕ, then Vðγ, pÞ = cesððp n ÞÞ examined by Sanhan and Suantai [21] (2) If γ i = i + 1 and p i = p, for all i ∈ ℕ, then Vðγ, pÞ = ces p . Some authors [22][23][24] investigated various sorts of Cesáro summable sequence spaces Definition 5 (see [25]). A class of linear sequence spaces X is called a special space of sequences (sss) if…”

“…Definition 7 (see [25]). A subclass of (sss) is called a premodular (sss) if there is a function σ : X ⟶ ½0, ∞Þ verifying the conditions (i) σðvÞ ≥ 0 for all v ∈ X and σðvÞ = 0 ⇔ v = θ, here θ is the zero element of X (ii) there is L ≥ 1 with σðηvÞ ≤ L|η|σðvÞ for each v ∈ X, and η ∈ ℂ…”