Research Article Mappings of Type Special Space of Sequences

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Research Article

Mappings of Type Special Space of Sequences

Awad A. Bakery

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1_{Department of Mathematics, Faculty of Science and Arts, University of Jeddah (UJ), P.O. Box 355, Khulais 21921, Saudi Arabia} 2_{Department of Mathematics, Faculty of Science, Ain Shams University, P.O. Box 1156, Abbassia, Cairo 11566, Egypt}

Correspondence should be addressed to Awad A. Bakery; awad [email protected] Received 12 April 2016; Revised 26 June 2016; Accepted 28 July 2016

Academic Editor: Gianluca Vinti

Copyright © 2016 Awad A. Bakery. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We give sufficient conditions on a special space of sequences defined by Mohamed and Bakery (2013) such that the finite rank operators are dense in the complete space of operators whose approximation numbers belong to this sequence space. Hence, under a few conditions, every compact operator would be approximated by finite rank operators. We apply it on the sequence space defined

by Tripathy and Mahanta (2003). Our results match those known for𝑝-absolutely summable sequences of reals.

1. Introduction and Basic Definitions

By 𝜔 and 𝐿(𝑉, 𝑊), we will denote the spaces of all real sequences and all bounded linear operators between two Banach spaces𝑉into𝑊, respectively. In [1], Pietsch, by using the approximation numbers and p-absolutely summable sequences of real numbers, formed the operator ideals. In [2], Mohamed and Bakery have considered the spaceℓ_𝑀, when

𝑀(𝑡) = 𝑡𝑝 _{(0 < 𝑝 < ∞)}_{, which matches especially} _ℓ𝑝_.

A subclass𝑈 of𝐿 = {𝐿(𝑉, 𝑊)} is an operator ideal if its components verify the following conditions:

(i) The space 𝐹(𝑉, 𝑊)of all finite rank operators is a subset of𝑈(𝑉, 𝑊).

(ii) The space𝑈(𝑉, 𝑊)is linear.

(iii) For two Banach spaces𝑉₀and𝑊₀, if𝑇 ∈ 𝐿(𝑉₀, 𝑉),𝑆 ∈

𝑈(𝑉, 𝑊), and𝑅 ∈ 𝐿(𝑊, 𝑊₀), then𝑅𝑆𝑇 ∈ 𝑈(𝑉₀, 𝑊₀). See [3, 4].

An Orlicz function is a function 𝑀 : [0, ∞) → [0, ∞) which is convex, positive, nondecreasing, and continuous, where𝑀(0) = 0and lim_𝑥→∞𝑀(𝑥) = ∞. An Orlicz function

𝑀is said to satisfy Δ₂-condition for all values of𝑥 ≥ 0if there exists a constant𝑘 > 0, such that𝑀(2𝑥) ≤ 𝑘𝑀(𝑥).

Lindenstrauss and Tzafriri [5] used the idea of an Orlicz function to define Orlicz sequence spaces as follows:

ℓ_𝑀= {𝑥 ∈ 𝜔 : ∃𝜆 > 0with𝜌 (𝜆𝑥) = ∞ ∑ 𝑘=1 𝑀 (|𝜆𝑥 (𝑘)|) < ∞} . (1)

(ℓ_𝑀, ‖𝑥‖)is a Banach space, where ‖𝑥‖ = inf{𝜆 > 0 :

𝜌(𝑥/𝜆) ≤ 1}.The spaceℓ𝑝is an Orlicz sequence space with

𝑀(𝑥) = 𝑥𝑝_for_{1 ≤ 𝑝 < ∞}_.

Remark 1. For any Orlicz function 𝑀, we have 𝑀(𝜆𝑥) ≤

𝜆𝑀(𝑥), for all𝜆with0 < 𝜆 < 1.

Let𝑃_𝑠be the class of all subsets ofN= {0, 1, 2, . . .}that do not contain more than𝑠number of elements and let{𝜙_𝑛}be a nondecreasing sequence of positive reals such that𝑛𝜙_𝑛+1≤

(𝑛+1)𝜙_𝑛, for all𝑛 ∈N. Tripathy and Mahanta [6] defined and studied the following sequence space:

𝑚 (𝜙 , 𝑀) = {𝑥 = (𝑥_𝑘) ∈ 𝜔 : ∃𝜁 > 0with sup 𝑠≥1,𝜎∈𝑃𝑠 1 𝜙_𝑠 _𝑘∈𝜎∑𝑀 (󵄨󵄨󵄨󵄨𝑥𝜁𝑘󵄨󵄨󵄨󵄨) < ∞} , (2)

Volume 2016, Article ID 4372471, 6 pages http://dx.doi.org/10.1155/2016/4372471

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with the norm 𝜌 (𝑥) =inf {𝜁 > 0 : sup 𝑠≥1,𝜎∈𝑃𝑠 1 𝜙_𝑠_𝑘∈𝜎∑𝑀 (󵄨󵄨󵄨󵄨𝑥𝜁𝑘󵄨󵄨󵄨󵄨) ≤ 1} . (3) Lemma 2. (i)ℓ_𝑀⊆ 𝑚(𝜙, 𝑀).

(ii)ℓ_𝑀= 𝑚(𝜙, 𝑀)if and only ifsup_𝑠𝜙_𝑠< ∞.

As of late, different classes of sequences have been pre-sented using Orlicz functions by Braha [7], Raj and Sharma Sunil [8], Raj et al. [9], and many others ([10–13]).

Definition 3(see [14, 15]). A special space of sequences (sss) is a linear space𝐸with the following:

(1)𝑒_𝑛 ∈ 𝐸for all𝑛 ∈N, where𝑒_𝑛 = {0, 0, . . . , 1, 0, 0, . . .} with 1 appearing at𝑛th place for all𝑛 ∈N.

(2)𝐸is solid󸀠󸀠.

(3)(𝑥₀, 𝑥₀, 𝑥₁, 𝑥₁, . . .) ∈ 𝐸, if(𝑥_𝑛)_𝑛∈N∈ 𝐸.

A premodular (sss)𝐸_𝜌is a (sss) and there is a function𝜌 :

𝐸 → [0, ∞[with the following:

(i)𝜌(𝑥) ≥ 0, for each𝑥 ∈ 𝐸and𝜌(𝑥) = 0 ⇔ 𝑥 = 𝜃, where𝜃is the zero element of𝐸.

(ii)𝜌satisfiesΔ₂-condition.

(iii) For each𝑥, 𝑦 ∈ 𝐸,𝜌(𝑥 + 𝑦) ≤ 𝑘(𝜌(𝑥) + 𝜌(𝑦))holds for some𝑘 ≥ 1.

(iv) The space 𝐸 is 𝜌-solid; that is,𝜌((𝑥_𝑛)) ≤ 𝜌((𝑦_𝑛)), whenever|𝑥_𝑛| ≤ |𝑦_𝑛|, for all𝑛 ∈N.

(v) For some numbers𝑘₀≥ 1, the inequality𝜌((𝑥_𝑛)_𝑛∈N) ≤

𝜌((𝑥₀, 𝑥₀, 𝑥₁, 𝑥₁, . . .)) ≤ 𝑘₀𝜌((𝑥_𝑛)_𝑛∈N)holds.

(vi)𝐹 = 𝐸_𝜌; that is, the set of all finite sequences𝐹is𝜌 -dense in𝐸.

(vii) For each𝜆 > 0, there is a constant𝜉 > 0such that

𝜌(𝜆, 0, 0, 0, . . .) ≥ 𝜉𝜆𝜌(1, 0, 0, 0, . . .).

Condition (ii) says that𝜌is continuous at𝜃. The function

𝜌defines a metrizable topology in𝐸and the linear space𝐸 enriched with this topology is denoted by𝐸_𝜌.

Definition 4(see [16]). Consider the following:

𝑈_𝐸appfl{𝑈_𝐸app(𝑉, 𝑊)} , (4) where

𝑈_𝐸app(𝑉, 𝑊)fl{𝑇 ∈ 𝐿 (𝑉, 𝑊) : (𝛼𝑛(𝑇))𝑛∈N ∈ 𝐸} . (5)

Theorem 5 (see [2]). If𝐸is a (sss), then𝑈_𝐸𝑎𝑝𝑝is an operator

ideal.

We explain some results related to the operator spaces.

2. Main Results

In this part, we give sufficient conditions on 𝐸 such that the finite rank operators are dense in the complete space of operators𝑈_𝐸app(𝑉, 𝑊).

Lemma 6. If𝐸_𝜌 is a premodular (sss) and (𝑥_𝑛) ∈ 𝐸_𝜌 is a

decreasing sequence of positive reals, then

𝜌 (⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0, 𝑥2𝑛 _𝑛, 𝑥_𝑛+1, 𝑥_𝑛+2, . . .) ≤ 𝑘₀𝜌 (⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0, 𝑥𝑛 _𝑛, 𝑥_𝑛+1, 𝑥_𝑛+2, . . .) .

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Proof. By using Definition 3, conditions (iv) and (v), and since the elements of𝐸are decreasing, we get

𝜌 (⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0, 𝑥2𝑛 _𝑛, 𝑥_𝑛+1, 𝑥_𝑛+2, . . .)

≤ 𝜌 (⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0, 𝑥2𝑛 _𝑛, 𝑥_𝑛, 𝑥_𝑛+1, 𝑥_𝑛+1, . . .) ≤ 𝑘₀𝜌 (⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0, 𝑥𝑛 _𝑛, 𝑥_𝑛+1, 𝑥_𝑛+2, . . .) .

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Theorem 7. Let𝐸_𝜌be a premodular (sss); then𝐹(𝑉, 𝑊)𝑔 =

𝑈_𝐸𝜌𝑎𝑝𝑝(𝑉, 𝑊), where𝑔(𝑇) = 𝜌(𝛼_𝑛(𝑇)_𝑛∈N).

Proof. To prove that𝐹(𝑉, 𝑊) ⊆ 𝑈_𝐸app(𝑉, 𝑊), since𝑒_𝑚 ∈ 𝐸for each𝑚 ∈ N, from the linearity of𝐸and𝑇 ∈ 𝐹(𝑉, 𝑊), then finitely many elements of(𝛼_𝑛(𝑇))_𝑛∈Nare different from zero. Hence,𝑇 ∈ 𝑈_𝐸app(𝑉, 𝑊). For the other inclusion𝑈_𝐸app(𝑉, 𝑊) ⊆

𝐹(𝑉, 𝑊), let 𝑇 ∈ 𝑈_𝐸app(𝑉, 𝑊) and, from the definition of approximation numbers, there is𝑁 ∈ N,𝑁 > 0,𝐴_𝑁with rank(𝐴_𝑁) ≤ 𝑁and also

󵄩󵄩󵄩󵄩𝑇 − 𝐴𝑁󵄩󵄩󵄩󵄩 ≤ 2𝛼𝑁(𝑇) . (8)

Since𝛼_𝑁(𝑇) → 0as𝑁 → ∞, then ‖𝑇 − 𝐴_𝑁‖ → 0as

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𝑁 → ∞, by taking𝑁 = 8𝜂, where𝜂is a natural number. From Definition 3, condition (iii), we have

𝑑 (𝑇, 𝐴𝑁) = 𝜌 ((𝛼𝑛(𝑇 − 𝐴𝑁))𝑛∈N) = 𝜌 [(𝛼0(𝑇 − 𝐴𝑁) , 𝛼1(𝑇 − 𝐴𝑁) , . . . , 𝛼8𝜂−1(𝑇 − 𝐴𝑁) , 0, 0, 0, . . .) + ( 8𝜂 ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ 0, 0, 0, . . . , 0, 𝛼8𝜂(𝑇 − 𝐴𝑁) , 𝛼8𝜂+1(𝑇 − 𝐴𝑁) , . . . , 𝛼12𝜂−1(𝑇 − 𝐴𝑁) , 0, 0, 0, . . .) + (⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0, 𝛼12𝜂 12𝜂(𝑇 − 𝐴𝑁) , 𝛼12𝜂+1(𝑇 − 𝐴𝑁) , . . .)] ≤ 𝑘2[𝜌 (𝛼0(𝑇 − 𝐴𝑁) , 𝛼1(𝑇 − 𝐴𝑁) , . . . , 𝛼8𝜂−1(𝑇 − 𝐴𝑁) , 0, 0, 0, . . .) + 𝜌 (⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0, 𝛼8𝜂 8𝜂(𝑇 − 𝐴𝑁) , 𝛼8𝜂+1(𝑇 − 𝐴𝑁) , . . . , 𝛼12𝜂−1(𝑇 − 𝐴𝑁) , 0, 0, 0, . . .) + 𝜌 ( 12𝜂 ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ 0, 0, 0, . . . , 0, 𝛼12𝜂(𝑇 − 𝐴𝑁) , 𝛼12𝜂+1(𝑇 − 𝐴𝑁) , . . .)] = 𝑘2[𝐼1(𝑁) + 𝐼2(𝜂) + 𝐼3(𝜂)] . (9)

Since𝛼_𝑛(𝐴_𝑁) = 0for𝑛 ≥ 𝑁, then

𝛼𝑛(𝑇 − 𝐴𝑁) ≤ 𝛼𝑛−𝑁(𝑇) . (10)

By using Lemma 6, inequality (10), and Definition 3, condi-tion (v), we get 𝐼₃(𝜂) = 𝜌 (⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0, 𝛼12𝜂 _12𝜂(𝑇 − 𝐴_𝑁) , 𝛼_12𝜂+1(𝑇 − 𝐴_𝑁) , . . .) ≤ 𝜌 (⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0, 𝛼12𝜂 _4𝜂(𝑇) , 𝛼4𝜂+1(𝑇) , . . .) ≤ 𝑘0𝜌 ( 6𝜂 ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ 0, 0, 0, . . . , 0, 𝛼4𝜂(𝑇) , 𝛼_4𝜂+1(𝑇) , . . .) ≤ 𝑘2₀𝜌 (⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0, 𝛼3𝜂 _4𝜂(𝑇) , 𝛼_4𝜂+1(𝑇) , . . .) ≤ 𝑘2₀𝜌 (⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0, 𝛼3𝜂 _3𝜂(𝑇) , 𝛼3𝜂+1(𝑇) , . . .) = 𝑘20𝜌 ((𝛼𝑛(𝑇))∞𝑛=3𝜂) 󳨀→ 0 as𝜂 󳨀→ ∞. (11)

Now, using Lemma 6 and Definition 3, condition (v), we have

𝐼₂(𝜂) = 𝜌 (⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0, 𝛼8𝜂 _8𝜂(𝑇 − 𝐴_𝑁) , 𝛼_8𝜂+1(𝑇 − 𝐴_𝑁) , . . . , 𝛼_12𝜂−1(𝑇 − 𝐴_𝑁) , 0, 0, 0, . . .) ≤ 𝑘₀𝜌 (⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0, 𝛼4𝜂 _8𝜂(𝑇 − 𝐴_𝑁) , 𝛼_8𝜂+1(𝑇 − 𝐴_𝑁) , . . . , 𝛼_12𝜂−1(𝑇 − 𝐴_𝑁) , 0, 0, 0, . . .) ≤ 𝑘₀𝜌 (𝛼₀(𝑇 − 𝐴_𝑁) , 𝛼₁(𝑇 − 𝐴_𝑁) , . . . , 𝛼8𝜂−1(𝑇 − 𝐴𝑁) , 0, 0, 0, . . .) = 𝑘0𝐼1(𝑁) . (12)

Finally, we have to show that𝐼₁(𝑁) → 0as𝑁 → ∞. Since𝑇 ∈

𝑈_𝐸app(𝑉, 𝑊)and𝜌is continuous at𝜃, we have𝜌(𝛼_𝑘(𝑇))∞_𝑘=𝑛→ 0 as𝑛 → ∞. Then, for each𝜀 > 0, there exists𝑁₀(𝜀)such that for all𝑛 ≥ 𝑁₀(𝜀)we have

𝜌 ((𝛼_𝑘(𝑇))∞_𝑘=𝑛) < 𝜀. (13) By taking𝜀₁= 𝜀/3𝑙𝑘for each𝑛 ≥ 𝑁₀(𝜀₁)and using inequality (13) and Definition 3, conditions (ii) and (iii), then we have

𝐼₁(𝑁) = 𝜌 (𝛼0(𝑇 − 𝐴𝑁) , 𝛼1(𝑇 − 𝐴𝑁) , . . . , 𝛼𝑁−1(𝑇 − 𝐴_𝑁) , 0, 0, 0, . . .) ≤ 𝑘 [𝜌 (𝛼₀(𝑇 − 𝐴_𝑁) , 𝛼1(𝑇 − 𝐴𝑁) , . . . , 𝛼𝑁0−1(𝑇 − 𝐴𝑁) , 0, 0, 0, . . .) + 𝜌 (⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0, 𝛼𝑁0 _𝑁₀(𝑇 − 𝐴_𝑁) , 𝛼_𝑁0+1(𝑇 − 𝐴_𝑁) , . . . , 𝛼𝑁−1(𝑇 − 𝐴𝑁) , 0, 0, 0, . . .)] ≤ 𝑘 [𝜌 (󵄩󵄩󵄩󵄩𝑇 − 𝐴𝑁󵄩󵄩󵄩󵄩, 󵄩󵄩󵄩󵄩𝑇 − 𝐴𝑁󵄩󵄩󵄩󵄩,...,󵄩󵄩󵄩󵄩𝑇 − 𝐴𝑁󵄩󵄩󵄩󵄩,0,0,0,...) + 𝜌 (⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0, 2𝛼𝑁0 _𝑁(𝑇) , 2𝛼_𝑁(𝑇) , . . . , 2𝛼_𝑁(𝑇) , 0, 0, 0, . . .)] ≤ 𝑘 [󵄩󵄩󵄩󵄩𝑇 − 𝐴𝑁󵄩󵄩󵄩󵄩𝑙𝜌( 𝑁0 ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ 1, 1, 1, . . . , 1, 0, 0, 0, . . .) + 2𝑙𝜌 (⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, 0, 0, . . . , 0, 𝛼𝑁0 _𝑁0(𝑇) , 𝛼_𝑁0+1(𝑇) , . . . , 𝛼𝑁(𝑇) , 0, 0, 0, . . .)] ≤ 𝑘 [󵄩󵄩󵄩󵄩𝑇 − 𝐴𝑁󵄩󵄩󵄩󵄩𝑙𝑘1(𝜀) + 2𝑙𝜀1] , (14) where 𝑘₁(𝜀) = 𝜌( 𝑁0 ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ 1, 1, 1, . . . , 1, 0, 0, 0, . . .), and since ‖𝑇 −

(4)

such that ‖𝑇 − 𝐴_𝑁‖𝑘₁(𝜀) ≤ 𝜀₁, for that we have 𝐼₁(𝑁) ≤

𝑘[𝑙𝜀₁+ 2𝑙𝜀₁] = 3𝑘𝑙𝜀₁= 𝜀.This completes the proof.

We give here the sufficient conditions on the sequence spaces𝑚(𝜙, 𝑀)such that the class of all bounded linear oper-ators between any arbitrary Banach spaces with (𝛼_𝑛(𝑇))_𝑛∈Nin these sequence spaces form an ideal operator; the ideal of the finite rank operators in the class of Banach spaces is dense in

𝑈_{𝑚(𝜙,𝑀)}app (𝑉, 𝑊).

Theorem 8. Let 𝑀 be an Orlicz function satisfying Δ₂

-condition. Then,

(a)𝑈_{𝑚(𝜙,𝑀)}𝑎𝑝𝑝 is an operator ideal,

(b)𝐹(𝑉, 𝑊) = 𝑈_{𝑚(𝜙,𝑀)}𝑎𝑝𝑝 (𝑉, 𝑊).

Proof. We first prove that the space𝑚(𝜙, 𝑀)is a (sss). (1) Let𝜆₁, 𝜆₂ ∈ Rand𝑥, 𝑦 ∈ 𝑚(𝜙, 𝑀); then there exist

𝜁₁> 0and𝜁₂> 0such that sup 𝑠≥1,𝜎∈𝑃𝑠 1 𝜙𝑠𝑘∈𝜎∑ 𝑀 (󵄨󵄨󵄨󵄨𝑥𝑘󵄨󵄨󵄨󵄨 𝜁1 ) < ∞, sup 𝑠≥1,𝜎∈𝑃𝑠 1 𝜙_𝑠_𝑘∈𝜎∑𝑀 (󵄨󵄨󵄨󵄨𝑥𝜁₂𝑘󵄨󵄨󵄨󵄨) < ∞. (15)

Let𝜁₃ = max(2|𝜆₁|𝜁₁, 2|𝜆₂|𝜁₂). Since𝑀is nondecreasing convex function withΔ₂-condition, we have

∑ 𝑘∈𝜎 𝑀 (󵄨󵄨󵄨󵄨𝜆1𝑥𝑘_𝜁+ 𝜆2𝑦𝑘󵄨󵄨󵄨󵄨 3 ) ≤ 1₂[∑ 𝑘∈𝜎 𝑀 (󵄨󵄨󵄨󵄨𝑥_𝜁𝑘󵄨󵄨󵄨󵄨 1 ) + ∑𝑘∈𝜎 𝑀 (󵄨󵄨󵄨󵄨𝑦_𝜁𝑘󵄨󵄨󵄨󵄨 2 )] . (16) So, we get sup 𝑠≥1,𝜎∈𝑃𝑠 1 𝜙_𝑠_𝑘∈𝜎∑𝑀 (󵄨󵄨󵄨󵄨𝜆1 𝑥_𝑘+ 𝜆₂𝑦_𝑘󵄨󵄨󵄨󵄨 𝜁₃ ) ≤ 1₂[ sup 𝑠≥1,𝜎∈𝑃𝑠 1 𝜙_𝑠_𝑘∈𝜎∑𝑀 (󵄨󵄨󵄨󵄨𝑥𝜁𝑘₁󵄨󵄨󵄨󵄨) + sup 𝑠≥1,𝜎∈𝑃𝑠 1 𝜙_𝑠_𝑘∈𝜎∑𝑀 (󵄨󵄨󵄨󵄨𝑦𝜁₂𝑘󵄨󵄨󵄨󵄨)] . (17)

Thus,𝜆₁𝑥 + 𝜆₂𝑦 ∈ 𝑚(𝜙 , 𝑀). Hence,𝑚(𝜙, 𝑀)is a linear space over the field of real numbers. Also, since𝑒_𝑛 ∈ ℓ_𝑀andℓ_𝑀⊆

𝑚(𝜙, 𝑀), we have𝑒_𝑛∈ 𝑚(𝜙, 𝑀)for all𝑛 ∈N.

(2) Let𝑥 ∈ 𝜔and𝑦 = (𝑦_𝑘)∞_𝑘=0∈ 𝑚(𝜙, 𝑀)with|𝑥_𝑘| ≤ |𝑦_𝑘| for each𝑘 ∈N; since𝑀is nondecreasing, then we get

sup 𝑠≥1,𝜎∈𝑃𝑠 1 𝜙_𝑠_𝑘∈𝜎∑𝑀 (󵄨󵄨󵄨󵄨𝑥𝜁𝑘󵄨󵄨󵄨󵄨) ≤𝑠≥1,𝜎∈𝑃𝑠sup 1 𝜙_𝑠_𝑘∈𝜎∑𝑀 (󵄨󵄨󵄨󵄨𝑦𝜁𝑘󵄨󵄨󵄨󵄨) < ∞; (18) then𝑥 = (𝑥_𝑘)∞_𝑘=0∈ 𝑚(𝜙, 𝑀).

(3) Let𝑥 = (𝑥_𝑘)∞_𝑘=0∈ 𝑚(𝜙, 𝑀); then we have

sup 𝑠≥1,𝜎∈𝑃𝑠 1 𝜙𝑠𝑘∈𝜎∑ 𝑀 (󵄨󵄨󵄨󵄨𝑥[𝑘/2]󵄨󵄨󵄨󵄨 𝜁 ) ≤ 2 sup 𝑠≥1,𝜎∈𝑃𝑠 1 𝜙_𝑠_𝑘∈𝜎∑𝑀 (󵄨󵄨󵄨󵄨𝑥𝜁𝑘󵄨󵄨󵄨󵄨) < ∞; (19) then𝑥 = (𝑥_[𝑘/2])∞_𝑘=0∈ 𝑚(𝜙, 𝑀).

Finally, we have proved that the space𝑚(𝜙, 𝑀)with𝜌(𝑥) is a premodular (sss).

(i) Clearly,𝜌(𝑥) ≥ 0for all𝑥 ∈ 𝑚(𝜙 , 𝑀)and𝜌(𝑥) = 0 ⇔

𝑥 = 𝜃.

(ii) Let𝜆 ∈Rand𝑥 ∈ 𝑚(𝜙 , 𝑀); then for𝜆 ̸= 0we have

𝜌 (𝜆𝑥) =inf {𝜁 > 0 : sup 𝑠≥1,𝜎∈𝑃𝑠 1 𝜙_𝑠_𝑘∈𝜎∑𝑀 (󵄨󵄨󵄨󵄨𝜆𝑥𝜁𝑘󵄨󵄨󵄨󵄨) ≤ 1} =inf {|𝜆| 𝜇 > 0 : sup 𝑠≥1,𝜎∈𝑃𝑠 1 𝜙_𝑠_𝑘∈𝜎∑𝑀 (󵄨󵄨󵄨󵄨𝑥𝜇𝑘󵄨󵄨󵄨󵄨) ≤ 1} , (20)

where 𝜇 = 𝜁/|𝜆|. Thus, 𝜌(𝜆𝑥) = |𝜆|inf{𝜇 > 0 :

sup_{𝑠≥1,𝜎∈𝑃}_𝑠(1/𝜙_𝑠) ∑_𝑘∈𝜎𝑀(|𝑥_𝑘|/𝜇) ≤ 1} = |𝜆|𝜌(𝑥).

Also, for𝜆 = 0, we have𝜌(𝜆𝑥) = 𝜆𝜌(𝑥) = 0.

(iii) Let𝑥, 𝑦 ∈ 𝑚(𝜙, 𝑀); then there exist𝜁₁> 0and𝜁₂> 0 such that sup 𝑠≥1,𝜎∈𝑃𝑠 1 𝜙_𝑠_𝑘∈𝜎∑𝑀 (󵄨󵄨󵄨󵄨𝑥𝜁₁𝑘󵄨󵄨󵄨󵄨) ≤ 1, sup 𝑠≥1,𝜎∈𝑃𝑠 1 𝜙_𝑠_𝑘∈𝜎∑𝑀 (󵄨󵄨󵄨󵄨𝑥𝜁₂𝑘󵄨󵄨󵄨󵄨) ≤ 1. (21)

Let𝜁 = 𝜁₁+ 𝜁₂, and since𝑀is nondecreasing and convex, then we have sup 𝑠≥1,𝜎∈𝑃𝑠 1 𝜙_𝑠_𝑘∈𝜎∑𝑀 (󵄨󵄨󵄨󵄨𝑥𝑘 + 𝑦_𝑘󵄨󵄨󵄨󵄨 𝜁 ) ≤𝑠≥1,𝜎∈𝑃𝑠sup 1 𝜙_𝑠 ⋅ ∑ 𝑘∈𝜎 𝑀 (󵄨󵄨󵄨󵄨𝑥𝑘󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑦𝑘󵄨󵄨󵄨󵄨 𝜁₁+ 𝜁₂ ) ≤𝑠≥1,𝜎∈𝑃sup𝑠 1 𝜙_𝑠 ⋅ ∑ 𝑘∈𝜎 [( 𝜁1 𝜁₁+ 𝜁₂) 𝑀 (󵄨󵄨󵄨󵄨𝑥𝜁₁𝑘󵄨󵄨󵄨󵄨) + ( 𝜁2 𝜁₁+ 𝜁₂) 𝑀 (󵄨󵄨󵄨󵄨𝑦𝜁₂𝑘󵄨󵄨󵄨󵄨)] ≤ ( 𝜁₁ 𝜁₁+ 𝜁₂)𝑠≥1,𝜎∈𝑃𝑠sup 1 𝜙_𝑠 ⋅ ∑ 𝑘∈𝜎 𝑀 (󵄨󵄨󵄨󵄨𝑥𝑘󵄨󵄨󵄨󵄨 𝜁₁ ) + ( 𝜁₂ 𝜁₁+ 𝜁₂) ∑_𝑘∈𝜎𝑀 (󵄨󵄨󵄨󵄨𝑦𝜁₂𝑘󵄨󵄨󵄨󵄨) ≤ 1. (22)

(5)

Since𝜁’s are nonnegative, we have 𝜌 (𝑥 + 𝑦) =inf {𝜁 > 0 : sup 𝑠≥1,𝜎∈𝑃𝑠 1 𝜙_𝑠_𝑘∈𝜎∑𝑀 (󵄨󵄨󵄨󵄨𝑥𝑘 + 𝑦_𝑘󵄨󵄨󵄨󵄨 𝜁 ) ≤ 1} ≤inf {𝜁₁> 0 : sup 𝑠≥1,𝜎∈𝑃𝑠 1 𝜙_𝑠_𝑘∈𝜎∑𝑀 (󵄨󵄨󵄨󵄨𝑥𝜁₁𝑘󵄨󵄨󵄨󵄨) ≤ 1} +inf {𝜁₂> 0 : sup 𝑠≥1,𝜎∈𝑃𝑠 1 𝜙_𝑠_𝑘∈𝜎∑𝑀 (󵄨󵄨󵄨󵄨𝑦𝜁₂𝑘󵄨󵄨󵄨󵄨) ≤ 1} = 𝜌 (𝑥) + 𝜌 (𝑦) . (23)

(iv) Let |𝑥_𝑘| ≤ |𝑦_𝑘| for each 𝑘 ∈ N, and since 𝑀 is nondecreasing, then we get

sup 𝑠≥1,𝜎∈𝑃𝑠 1 𝜙_𝑠_𝑘∈𝜎∑𝑀 (󵄨󵄨󵄨󵄨𝑥𝜁𝑘󵄨󵄨󵄨󵄨) ≤𝑠≥1,𝜎∈𝑃sup𝑠 1 𝜙_𝑠_𝑘∈𝜎∑𝑀 (󵄨󵄨󵄨󵄨𝑦𝜁𝑘󵄨󵄨󵄨󵄨) ; (24) thus, inf {𝜁 > 0 : sup 𝑠≥1,𝜎∈𝑃𝑠 1 𝜙_𝑠_𝑘∈𝜎∑𝑀 (󵄨󵄨󵄨󵄨𝑥𝜁𝑘󵄨󵄨󵄨󵄨)} ≤inf {𝜁 > 0 : sup 𝑠≥1,𝜎∈𝑃𝑠 1 𝜙𝑠𝑘∈𝜎∑ 𝑀 (󵄨󵄨󵄨󵄨𝑦𝑘󵄨󵄨󵄨󵄨 𝜁 )} . (25) So,𝜌(𝑥) ≤ 𝜌(𝑦). (v) Since sup 𝑠≥1,𝜎∈𝑃𝑠 1 𝜙_𝑠_𝑘∈𝜎∑𝑀 (󵄨󵄨󵄨󵄨𝑥[𝑘/2]𝜁 󵄨󵄨󵄨󵄨) ≤ 2 sup 𝑠≥1,𝜎∈𝑃𝑠 1 𝜙_𝑠 _𝑘∈𝜎∑𝑀 (󵄨󵄨󵄨󵄨𝑥𝜁𝑘󵄨󵄨󵄨󵄨) , (26) we have inf {𝜁 > 0 : sup 𝑠≥1,𝜎∈𝑃𝑠 1 𝜙_𝑠_𝑘∈𝜎∑𝑀 (󵄨󵄨󵄨󵄨𝑥[𝑘/2]𝜁 󵄨󵄨󵄨󵄨)} ≤ 2inf {𝜁 > 0 : sup 𝑠≥1,𝜎∈𝑃𝑠 1 𝜙_𝑠_𝑘∈𝜎∑𝑀 (󵄨󵄨󵄨󵄨𝑥𝜁𝑘󵄨󵄨󵄨󵄨)} . (27) So,𝜌((𝑥_𝑘)) ≤ 𝜌((𝑥_[𝑘/2])) ≤ 2𝜌((𝑥_𝑘)).

(vi) For each𝑥 = (𝑥_𝑘)∞_𝑘=0∈ 𝑚(𝜙, 𝑀), then

𝜌 ((𝑥_𝑘)∞_𝑘=0) =inf {𝜁 > 0 : sup 𝑠≥1,𝜎∈𝑃𝑠 1 𝜙_𝑠_𝑘∈𝜎∑𝑀 (󵄨󵄨󵄨󵄨𝑥[𝑘/2]𝜁 󵄨󵄨󵄨󵄨) < ∞} ; (28)

we can find𝑡 ∈Nsuch that𝜌((𝑥_𝑘)∞_𝑘=𝑡) < ∞. This means the set of all finite sequences is𝜌-dense in𝑚(𝜙, 𝑀).

(vii) For any𝜆 > 0, there exists a constant𝜁 ∈ ]0, 1]such that

𝜌 (𝜆, 0, 0, 0, . . .) ≥ 𝜁𝜆𝜌 (1, 0, 0, 0, . . .) . (29) By using Theorems 5 and 7, the proof follows.

As special cases of the above theorem, we obtain the following corollaries.

Corollary 9. Ifsup_𝑠𝜙_𝑠< ∞, one gets that

(a)𝑈_ℓ𝑀𝑎𝑝𝑝is an operator ideal,

(b)𝐹(𝑉, 𝑊) = 𝑈_ℓ𝑎𝑝𝑝

𝑀 (𝑉, 𝑊).

Corollary 10. Ifsup_𝑠𝜙_𝑠< ∞and𝑀(𝑡) = 𝑡𝑝with0 < 𝑝 < ∞,

one gets that

(a)𝑈_ℓ𝑎𝑝𝑝𝑝 is an operator ideal,

(b)𝐹(𝑉, 𝑊) = 𝑈_ℓ𝑎𝑝𝑝𝑝 (𝑉, 𝑊). See [1].

Theorem 11. If𝐸_𝜌 is a premodular (sss), then𝑈_𝐸𝜌𝑎𝑝𝑝(𝑉, 𝑊)is

complete.

Proof. Let(𝑇_𝑚)be a Cauchy sequence in 𝑈_𝐸𝜌app(𝑉, 𝑊); then, by using Definition 3, condition (vii), and since𝑈_𝐸app

𝜌 (𝑉, 𝑊) ⊆ 𝐿(𝑉, 𝑊), we get 𝜌 ((𝛼_𝑛(𝑇_𝑖− 𝑇_𝑗))_𝑛∈N) ≥ 𝜌 (𝛼₀(𝑇_𝑖− 𝑇_𝑗) , 0, 0, 0, . . .) = 𝜌 (󵄩󵄩󵄩󵄩󵄩𝑇𝑖− 𝑇𝑗󵄩󵄩󵄩󵄩_{󵄩 , 0, 0, 0, . . .)} ≥ 𝜉 󵄩󵄩󵄩󵄩󵄩𝑇𝑖− 𝑇𝑗󵄩󵄩󵄩󵄩_{󵄩 𝜌 (1, 0, 0, 0, . . .) ;} (30)

then(𝑇_𝑚)is also a Cauchy sequence in 𝐿(𝑉, 𝑊). Since the space 𝐿(𝑉, 𝑊) is a Banach space, then there exists 𝑇 ∈

𝐿(𝑉, 𝑊)such that ‖𝑇_𝑚− 𝑇‖ → 0, as 𝑚 → ∞, and since

(𝛼_𝑛(𝑇_𝑚))_𝑛∈N ∈ 𝐸, for each𝑚 ∈ N, then from Definition 3, conditions (iii) and (v), and since𝜌is continuous at𝜃, we have

𝜌 ((𝛼_𝑛(𝑇))_𝑛∈N) = 𝜌 (𝛼_𝑛(𝑇 − 𝑇_𝑚+ 𝑇_𝑚))_𝑛∈N ≤ 𝑘𝜌 (𝛼_[𝑛/2](𝑇 − 𝑇_𝑚))_𝑛∈N + 𝑘𝜌 (𝛼_[𝑛/2](𝑇_𝑚))_𝑛∈N ≤ 𝑘𝜌 (󵄩󵄩󵄩󵄩𝑇𝑚− 𝑇󵄩󵄩󵄩󵄩)𝑛∈N + 𝑘𝜌 (𝛼_𝑛(𝑇_𝑚))_𝑛∈N< 𝜀; (31)

we get(𝛼_𝑛(𝑇))_𝑛∈N∈ 𝐸, and then𝑇 ∈ 𝑈_𝐸𝜌app(𝑉, 𝑊). This finishes the proof.

By applying Theorem 11 on 𝑚(𝜙, 𝑀), we can easily conclude the next corollaries.

Corollary 12. Pick up an Orlicz function𝑀which satisfies

Δ₂-condition. Then,𝑀is continuous from the right at0and

(6)

Corollary 13. Pick up an Orlicz function𝑀which satisfiesΔ₂ -condition withsup_𝑠𝜙_𝑠 < ∞. Then,𝑀is continuous from the right at0and𝑈_𝑙𝑀𝑎𝑝𝑝(𝑋, 𝑌)is complete.

Corollary 14. 𝑈_ℓ𝑎𝑝𝑝𝑝 (𝑉, 𝑊)is complete if𝑀(𝑡) = 𝑡𝑝and𝑝 ∈

(0, ∞).

Competing Interests

The author declares that he has no competing interests.

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