^{1, 2}

^{1}

^{2}

The aim of this paper is to give the sufficient conditions on the sequence space

Most of the operator ideals in the class of Banach spaces or in the class of normed spaces in linear functional analysis are defined by different scalar sequence spaces. In [

if

if

For a sequence

The space

If

(2) If

The idea of the paper is the following. We proceed in the following way: given a scalar sequence space

Throughout this paper, the sequence

the sequence

the sequence

Also, we define

Recently different classes of paranormed sequence spaces have been introduced and their different properties have been investigated by Et et al. [

The following well-known inequality will be used throughout the paper. For any bounded sequence of positive numbers

A class of linear sequence spaces

if

if

Let

the sequence

the sequence

We state the following result without proof.

We study here the operator ideals generated by the approximation numbers and the sequence space

(1-i) Let

(1-ii) Let

To prove that

(2) Let

(3) Let

Hence, from Theorem

The linear space

First, we show that every finite mapping

A subclass of the special space of sequences called premodular special space of sequences characterized for the existence of a function

there exists a constant

for some numbers

if

for some numbers

for each

for any

Let

If

If the following conditions are satisfied:

the sequence

the sequence

(i) Clearly,

(ii) Since

(iii) For some numbers

(iv) Let

(v) There exist some numbers

(vi) It is clear that the set of all finite sequences is

(vii) For any

Let

Let

Hence,

Let

Let

The author declares that there is no conflict of interests regarding the publication of this paper.

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. 857-007-D1434. The author, therefore, acknowledges with thanks DSR technical and financial support. Moreover, the author is most grateful to the editor and anonymous referee for careful reading of the paper and valuable suggestions which helped in improving an earlier version of it.