Journal of Function Spaces
Volume 2019 (2019), Article ID 7265010, 9 pages
https://doi.org/10.1155/2019/7265010

Research Article

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

Awad A. Bakery1,2 and Mustafa M. Mohammed1,3

¹Department of Mathematics, Faculty of Science and Arts, University of Jeddah(UJ), P.O. Box 355, Code 21921 Khulais, Saudi Arabia
²Department of Mathematics, Faculty of Science, Ain Shams University, P.O. Box 1156, Abbassia, Cairo 11566, Egypt
³Department of Statistics, Faculty of Science, Sudan University of Science & Technology, Khartoum, Sudan

Correspondence should be addressed to Mustafa M. Mohammed; mustasta@gmail.com

Received 11 February 2019; Accepted 15 April 2019; Published 2 May 2019

Guest Editor: Tuncer Acar

Copyright © 2019 Awad A. Bakery and Mustafa M. Mohammed. 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.

Abstract.

In this paper, we give the sufficient conditions on Orlicz-Cesáro mean sequence spaces , where is an Orlicz function such that the class of all bounded linear operators between arbitrary Banach spaces with its sequence of numbers which belong to forms an operator ideal. The completeness and denseness of its ideal components are specified and 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 such that the pre-quasi Banach operator ideal generated by approximation number is small. The above results coincide with that known for   .

1. Introduction

Throughout the paper, by , we mean the space of all real sequences, the real numbers, and and the space of all bounded linear operators from a normed space into a normed space . The operator ideals theory takes an importance in functional analysis, since it has numerous applications in fixed point theorem, geometry of Banach spaces, spectral theory, eigenvalue distributions theorem, etc. Some of the operator ideals in the class of normed spaces or Banach spaces in functional analysis are characterized by various scalar sequence spaces. For example the ideal of compact operators is defined by kolmogorov numbers and the space of convergent to zero sequences. Pietsch [1] inspected the operator ideals framed by the approximation numbers and the classical sequence space . He proved that the ideals of Hilbert Schmidt operators and nuclear operators between Hilbert spaces are defined by and , respectively, and the sequence of approximation numbers. In [2], Faried and Bakery examined the operator ideals developed by generalized Cesáro, Orlicz sequence spaces , and the approximation numbers. In [3], Faried and Bakery studied the operator ideals constructed by numbers, generalized Cesáro and Orlicz sequence spaces and show that the operator ideal formed by the previous sequence spaces and approximation numbers is small under certain conditions. Also summation process and sequences spaces applications are closely related to Korovkin type approximation theorems and linear positive operators studied by Costarelli and Vinti [4] and Altomare [5]. The idea of this paper is to examine a generalized class by using Orlicz-Cesáro mean sequence spaces and the sequence of -numbers, for which constructs an operator ideal. The components of as a pre-quasi Banach operator ideal containing finite dimensional operators as a dense subset and its completeness are proved. The inclusion relations between the pre-quasi operator ideals and the inclusion relations for their duals are determined. Finally, we show that the pre-quasi Banach operator ideal formed by the approximation numbers and is small under certain conditions. These results coincide with that known for , in [3]. Furthermore we give some examples which support our main results.

2. Definitions and Preliminaries

Definition 1 (see [6]). The sequence , for all is named an -function and the number is called the - number of if the following are satisfied:

(a) monotonicity: for all ;(b) additivity: for all , , ;(c) property of ideal: for all , , and , where and are normed spaces;(d) for every , ;(e) rank property: if then for every ;(f) property of norming:

 where is the identity operator on .

There are a few instances of -numbers; we notice the accompanying conditions:

(1)The n-th approximation number, denoted by , is defined by .(2)The n-th Hilbert number, denoted by , is defined by

(3)The n-th Weyl number, denoted by , is defined by

(4)The n-th Kolmogorov number, denoted by , is defined by

(5)The n-th Gel’fand number, denoted by , is defined by , where is a metric injection from the space to a higher space for an adequate index set . This number is independent of the choice of the higher space .(6)The n-th Chang number, denoted by , is defined by

Remark 2 (see [6]). Among all the -number sequences characterized above, it is easy to check that the approximation number, , is the largest and the Hilbert number, , is the smallest -number sequence, i.e., for any bounded linear operator . If is defined on a Hilbert space and compact, then all the -numbers correspond with the eigenvalues of , where .

Theorem 3 ([6], p.115). Let . Then

Theorem 4 ([6], p.90). An -number sequence is called injective if, for any metric injection , for all .

Theorem 5 ([6], p.95). An -number sequence is called surjective if, for any metric surjection , for all .

Theorem 6 ([6], pp.90-94). The Weyl and Gel’fand numbers are injective.

Theorem 7 ([6], pp.95). The Chang and Kolmogorov numbers are surjective.

Definition 8. A finite rank operator is a bounded linear operator whose dimension of the range space is finite.

Definition 9 ((dual -numbers) [7]). For each -number sequence , a dual -number function is defined by

where is the dual of .

Definition 10 ([8], p.152)). An -number sequence is called symmetric if for all . If , then the -number sequence is said to be completely symmetric.

Presently we express some known results of dual of an -number sequence.

Theorem 11 ([8], p.152). The approximation numbers are symmetric, i.e., for .

Remark 12 (see [9]). for every compact operator .

Theorem 13 ([8], p.153). Let . Then

In addition, if is a compact operator then .

Theorem 14 ([6], p.96). Let . Then

i.e., Chang numbers and Weyl numbers are dual to each other.

Theorem 15 ([8], p.153). The Hilbert numbers are completely symmetric, i.e., for all .

Definition 16 (see [10, 11]). The operator ideal is a subclass of linear bounded operators such that its components which are subsets of fulfill the accompanying conditions:

(i) where indicates one dimensional Banach space, where .(ii)For , then for any scalars .(iii)If , , and , then .

Definition 17 (see [12, 13]). An Orlicz function is a function , which is nondecreasing, convex, and continuous with and for and .

Definition 18. An Orlicz function is said to satisfy -condition for every values of , if there is , such that . The -condition is corresponding to for every values of and .

Lindenstrauss and Tzafriri [14] utilized the idea of an Olicz function to define Orlicz sequence space:

is a Banach space with the Luxemburg norm:

Every Orlicz sequence space contains a subspace that is isomorphic to , for some or ([15], Theorem 4.a.9).

In the recent past lot of work has been done on sequence spaces defined by Orlicz functions by Altin et al. [16], Et et al. ([17, 18]), Tripathy et al. ([19–21]), and Mohiuddine et al. ([22–25]).

Given an Orlicz function , the Orlicz-Cesáro mean sequence spaces is defined by

is a Banach space with the Luxemburg norm given by

It seems that Orlicz-Cesáro mean sequence spaces appeared for the first time in 1988, when Lim and Yee found their dual spaces [26]. Recently Cui, Hudzik, Petrot, Suantai, and Szymaszkiewicz obtained important properties of spaces [27]. In 2007 Maligranda, Petrot, and Suantai showed that is not B-convex, if and [28]. The extreme points and strong -points of have been characterized by Foralewski, Hudzik, and Szymaszkiewicz in [29]. In the case when , , the space is just a Cesáro sequence space , with the norm given by

It is well known that [30].

Definition 19 (see [31]). The Matuszewska Orlicz lower index of an Orlicz function is defined as follows:

Theorem 20 (see [31]). For any Orlicz function , we have if and only if . In particular, if then .

Theorem 21 (see [31]). Let and be Orlicz functions. If there exist such that and for all , then .

Theorem 22 (see [31]). Let and be Orlicz functions and , then if and only if there exist such that and for all .

Definition 23 (see [2]). A class of linear sequence spaces is called a special space of sequences (sss) having three properties:

(1) for all ,(2)if , and for every , then , “i.e., is solid”,(3)if , then , wherever means the integral part of .

Definition 24 (see [2]). A subclass of the special space of sequences is called a premodular (sss) if there is a function fulfilling the accompanying conditions:

(i) for each and , where is the zero element of ,(ii)there exists such that for all , and for any scalar ,(iii)for some , we have for every ,(iv)if for all , then ,(v)for some , we have

(vi)the set of all finite sequences is -dense in . This means for each and for each there exists such that ,(vii)there exists a constant such that for any .

We denote for the linear space equipped with the metrizable topology generated by .

Theorem 25 (see [32]). If , are infinite dimensional Banach spaces and is a monotonic decreasing sequence to zero, then there exists a bounded linear operator such that

Notations 26 (see [3]).

 , where . Also , where .

Theorem 27 (see [3]). If is a (sss), then is an operator ideal.

The concept of pre-quasi operator ideal which is more general than the usual classes of operator ideal.

Definition 28 (see [3]). A function is said to be a pre-quasi norm on the ideal fulfilling the accompanying conditions:

(1)for all , and if and only if ,(2)there exists a constant such that , for all and ,(3)there exists a constant such that , for all ,(4)there exists a constant such that if , , and , then , where and are normed spaces.

Theorem 29 (see [3]). Every quasi norm on the ideal is a pre-quasi norm on the ideal .

Here and after, we define where 1 appears at the place for all .

3. Main Results

We give here the conditions on Orlicz-Cesáro mean sequence spaces such that the class of all bounded linear operators between arbitrary Banach spaces with its sequence of numbers which belong to forms an operator ideal.

Theorem 30. If is an Orlicz function satisfying -condition and , then is an operator ideal.

Proof. (1-i) Let . Since is nondecreasing, convex, and satisfying -condition, we get for some that

then
(1-ii) Let and , and since is convex and satisfying -condition, we get for some that

then ; from (1-i) and (1-ii) is a linear space. Since , for all and , then from Theorem 20, we get , for all .
Let for all and ; since is nondecreasing, then we have

and we get .
Let . Since is satisfying -condition, we get for some that

then . Then is a (sss); hence by Theorem 27, is an operator ideal.

Corollary 31. is an operator ideal, if .

We give the conditions on Orlicz-Cesáro mean sequence spaces such that the ideal of the finite rank operators is dense in .

Theorem 32. , if is an Orlicz function satisfying -condition and .

Proof. Let us define on . First, we have to show that . Since , we have for each and is an Orlicz function satisfying -condition, so for each finite operator , i.e., we obtain which contains only finitely many terms different from zero; hence . Currently we prove that ; let ; we have ; and hence . By taking , hence there exists a such that for some , where . As is decreasing for every and is nondecreasing, we have

Hence, there exists such that rank and

Since is right continuous at 0 and nondecreasing, then on considering this

Let , and , since is Orlicz function and by using (22), (23), and (24), we have

Corollary 33. , if .

We express the accompanying theorem without verification; these can be set up utilizing standard procedure.

Theorem 34. The function is a pre-quasi norm on , if is an Orlicz function satisfying -condition and .

We give the sufficient conditions on Orlicz-Cesáro mean sequence spaces such that the components of the pre-quasi operator ideal are complete.

Theorem 35. If and are Banach spaces, is an Orlicz function satisfying -condition and , then is a pre-quasi Banach operator ideal.

Proof. Since is an Orlicz function satisfying -condition, then the function is a pre-quasi norm on . Let be a Cauchy sequence in . Since and , we can find a constant such that

then is also a Cauchy sequence in . While the space is a Banach space, there exists such that , while for every . Since is continuous at and for some , we obtain

we have , and then .

Corollary 36. If and are Banach spaces and , then is quasi Banach operator ideal, where .

Theorem 37. Let , be Orlicz functions and . For any infinite dimensional Banach spaces , and if there exist such that and for all , it is true that

Proof. Let and be infinite dimensional Banach spaces and there exist such that and for all ; if , then . From Theorems 21, 22, and 25, we have ; hence . It is easy to see that .

Corollary 38. For any infinite dimensional Banach spaces , , and , then .

We now study some properties of the pre-quasi Banach operator ideal .

Theorem 39. The pre-quasi Banach operator ideal (, ) is injective, if the -number sequence is injective.

Proof. Let and be any metric injection. Assume that , then Since the -number sequence is injective, we have , for all , . So . Hence and clearly is verified.

Remark 40. The pre-quasi Banach operator ideal (, ) and the pre-quasi Banach operator ideal (, ) are injective pre-quasi Banach operator ideal.

Theorem 41. The pre-quasi Banach operator ideal (, ) is surjective, if the -number sequence is surjective.

Proof. Let and be any metric surjection. Suppose that , then Since the -number sequence is surjective, we have , for all , . So . Hence and clearly is verified.

Remark 42. The pre-quasi Banach operator ideal (, ) and the pre-quasi Banach operator ideal (, ) are surjective pre-quasi Banach operator ideal.

Likewise, we have the accompanying inclusion relations between the pre-quasi Banach operator ideals.

Theorem 43. .
.

Proof. Since and for every and is nondecreasing, we obtain

Hence the result is as follows.

We presently express the dual of the pre-quasi operator ideal formed by different number sequences.

Theorem 44. The pre-quasi operator ideal is completely symmetric and the pre-quasi operator ideal is symmetric.

Proof. Since and , for all , we have and .

In perspective on Theorem 13, we express the following result without proof.

Theorem 45. The pre-quasi operator ideal and . In addition if is a compact operator from to , then .

In perspective on Theorem 14, we express the following result without proof.

Theorem 46. The pre-quasi operator ideal and .

Theorem 47. If is an Orlicz function satisfying -condition and , then the pre-quasi Banach operator ideal is small.

Proof. Since is an Orlicz function and , take . Then , where is a pre-quasi Banach operator ideal. Let and be any two Banach spaces. Assume that , then there exists a constant such that for all . Suppose that and are infinite dimensional Banach spaces. Then by Dvoretzky’s theorem [8] for , we have quotient spaces and subspaces of which can be mapped onto by isomorphisms and such that and . Consider be the identity map on , be the quotient map from onto , and be the natural embedding map from into . Let be the Bernstein numbers [7], then

for . Now since is nondecreasing and having -condition, we have

Therefore

for some . Thus we arrive at a contradiction since is arbitrary. Hence and both cannot be infinite dimensional when .

Theorem 48. If is an Orlicz function satisfying -condition and , then the pre-quasi Banach operator ideal is small.

Corollary 49. If , then the quasi Banach operator ideal is small.

Corollary 50. If , then the quasi Banach operator ideal is small.

4. Examples

We give some examples which support our main results.

Example 1. Let be an Orlicz function; the subspace of all order continuous elements of is defined as [27]

If is an Orlicz function satisfying -condition and , then the following conditions are satisfied:

(1) is an operator ideal.(2).(3)If and are Banach spaces, then is pre-quasi Banach operator ideal.(4)The pre-quasi Banach operator ideal is small.

Proof. Since is an Orlicz function satisfying -condition and , then from Theorem in [31] we have which completes the proof.

Example 2. Let be defined as

It is clear that is an Orlicz function and . Also is satisfying -condition since

Then the following conditions are satisfied:

(1) is an operator ideal.(2).(3)If and are Banach spaces, then is pre-quasi Banach operator ideal.(4)The pre-quasi Banach operator ideal is small.

In the following two examples we will explain the importance of the sufficient conditions.

Example 3. Let be defined as

It is clear that is an Orlicz function. Since , hence . The space is not operator ideal since . Also since is convex function and for , we have

for all , then . Although is satisfying -condition since

Example 4. Let , where is defined as

It is clear that is an Orlicz function. Let with for all . We have for that

Hence , so the space is not operator ideal and . Also since is convex function and for , we have

for all , then .

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Disclosure

The authors received no financial support for the research, authorship, and or publication of this article.

Conflicts of Interest

The authors declare that have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

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