Research

# Mappings of type Orlicz and generalized Cesáro sequence space

Nashat F Mohamed1* and Awad A Bakery12

Author Affiliations

1 Department of Mathematics, Faculty of Science, Ain Shams University, Cairo, Egypt

2 Department of Mathematics, Faculty of Science and Arts, King Abdulaziz University (KAU), P.O. Box 80200, Khulais, 21589, Saudi Arabia

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Journal of Inequalities and Applications 2013, 2013:186  doi:10.1186/1029-242X-2013-186

 Received: 7 January 2013 Accepted: 5 April 2013 Published: 18 April 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

We study the ideal of all bounded linear operators between any arbitrary Banach spaces whose sequence of approximation numbers belong to the generalized Cesáro sequence space and Orlicz sequence space , when , ; our results coincide with that known for the classical sequence space .

##### Keywords:
approximation numbers; operator ideal; generalized Cesáro sequence space; Orlicz sequence space

### 1 Introduction

By , we denote the space of all bounded linear operators from a normed space X into a normed space Y. The set of natural numbers will denote by and the real numbers by ℝ. By ω, we denote the space of all real sequences. A map which assigns to every operator a unique sequence is called an s-function and the number is called the nth s-numbers of T if the following conditions are satisfied:

(a) , for all .

(b) , for all .

(c) , for all , and .

(d) , for all , .

(e) If , for all .

(f) where is the identity operator on the Euclidean space . Example of s-numbers, we mention approximation number , Gelfand numbers , Kolmogorov numbers and Tichomirov numbers defined by: All of these numbers satisfy the following condition:

(I) .

(II) , where is a metric injection (a metric injection is a one to one operator with closed range and with norm equal one) from the space Y into a higher space for suitable index set Λ.

(III) .

(IV) .

(g) for all .

An operator ideal U is a subclass of such that its components satisfy the following conditions:

(i) , where K denotes the 1-dimensional Banach space, where .

(ii) If , then for any scalars , .

(iii) If , , then . See [1-3].

An Orlicz function is a function which is continuous, non-decreasing and convex with and for , and as . See [4,5].

If convexity of Orlicz function M is replaced by . Then this function is called modulus function, introduced by Nakano [6]; also, see [7,8] and [9]. An Orlicz function M is said to satisfy -condition for all values of u, if there exists a constant , such that (). The -condition is equivalent to for all values of u and for . Lindentrauss and Tzafriri [10] used the idea of Orlicz function to construct Orlicz sequence space

which is a Banach space with respect to the norm

For , the space coincides with the classical sequence space . Recently, different classes of sequences have been introduced by using an Orlicz function. See [11] and [12].

Remark 1.1 Let M be an Orlicz function then for all λ with .

For a sequence of positive real numbers with , for all the generalized Cesáro sequence space is defined by

where

The space is a Banach space with the norm

If is bounded, we can simply write

Also, some geometric properties of are studied by Sanhan and Suantai [13].

Throughout this paper, the sequence is a bounded sequence of positive real numbers, we denote where 1 appears at ith place for all . Different classes of paranormed sequence spaces have been introduced and their different properties have been investigated. See [14-18] and [19].

For any bounded sequence of positive numbers , we have the following well-known inequality , and for all . See [20].

### 2 Preliminary and notation

Definition 2.1 A class of linear sequence spaces E, called a special space of sequences (sss) having the following conditions:

(1) E is a linear space and , for each .

(2) If , and , for all , then i.e.E is solid’,

(3) if , then , where denotes the integral part of .

We call such space a pre modular special space of sequences if there exists a function , satisfies the following conditions:

(i) and , where θ is the zero element of E,

(ii) there exists a constant such that for all values of and for any scalar λ,

(iii) for some numbers , we have the inequality , for all ,

(iv) if , for all then ,

(v) for some numbers we have the inequality ,

(vi) for each there exists such that . This means the set of all finite sequences is ρ-dense in E.

(vii) for any there exists a constant such that .

It is clear that from condition (ii) that ρ is continuous at θ. The function ρ defines a metrizable topology in E endowed with this topology is denoted by .

Example 2.2 is a pre-modular special space of sequences for , with .

Example 2.3 is a pre-modular special space of sequences for , with .

Definition 2.4

where

### 3 Main results

Theorem 3.1is an operator ideal ifEis a special space of sequences (sss).

Proof To prove is an operator ideal:

(i) let and for all , since E is a linear space and for each , then ; for that , which implies .

(ii) Let and then from Definition 2.1 condition (3) we get and , since , is a decreasing sequence and from the definition of approximation numbers we get

Since E is a linear space and from Definition 2.1 condition (2) we get , hence .

(iii) If , and , then we get and since , from Definition 2.1 conditions (1) and (2) we get , then .

□

Theorem 3.2is an operator ideal, ifMis an Orlicz function satisfying-condition and there exists a constantsuch that.

Proof

(1-i) Let , since M is non-decreasing, we get , then .

(1-ii) , since M satisfies -condition, we get , for that , then from (1-i) and (1-ii) is a linear space over the field of numbers. Also for each since .

(2) Let for each , , since M is none decreasing, then we get , then .

(3) Let , , then . Hence, from Theorem 3.1, it follows that is an operator ideal.

□

Theorem 3.3is an operator ideal, ifis an increasing sequence of positive real numbers, and.

Proof

(1-i) Let since

then .

(1-ii) Let , , then

we get , from (1-i) and (1-ii) is a linear space.

To show that for each , since we have . Thus, we get

Hence .

(2) Let for each , then

since . Thus, .

(3) Let , then we have

Hence, . Hence, from Theorem 3.1 it follows that is an operator ideal.

□

Theorem 3.4LetMbe an Orlicz function. Then the linear spaceis dense in.

Proof Define on . First we prove that every finite mapping belongs to . Since for each and is a linear space then for every finite mapping the sequence contains only finitely many numbers different from zero. To prove that , let , we get , and since , let then there exists a natural number such that , since ρ is none decreasing and is decreasing for each , we get

then there exists , with , and by using the conditions of M we get

□

Corollary 3.5Ifand, we get. See[3].

Theorem 3.6The linear spaceis dense in, ifis an increasing sequence of positive real numbers withand.

Proof First we prove that every finite mapping belongs to . Since for each and is a linear space, then for every finite mapping i.e. the sequence contains only finitely many numbers different from zero. Now we prove that . Since , we have , let we get , and since , let then there exists a natural number such that for some , where , since is decreasing for each , we get

(1)

then there exists ,

(2)

and

(3)

since . Then there exists a natural number , with and . Since , then , so we can take

(4)

since is an increasing sequence and by using (1), (2), (3) and (4), we get

□

Theorem 3.7LetXbe a normed space, Ya Banach space andbe a pre modular special space of sequences (sss), thenis complete.

Proof Let be a Cauchy sequence in , then by using Definition 2.1 condition (vii) and since , we have

then is also Cauchy sequence in . Since the space is a Banach space, then there exists such that and since for all , ρ is continuous at θ and using Definition 2.1(iii), we have

Hence as such . □

Corollary 3.8LetXbe a normed space, Ya Banach space andMbe an Orlicz function such thatMsatisfies-condition. ThenMis continuous atandis complete.

Corollary 3.9LetXbe a normed space, Ya Banach space andbe an increasing sequence of positive real numbers withand, thenis complete.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

NFM gave the idea of the article. AAB carried out the proofs and its application. All authors read and approved the final manuscript.

### Acknowledgements

Dedicated to Professor Hari M Srivastava.

The authors wish to thank the referees for their careful reading of the paper and for their helpful suggestions.

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