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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 419560, 6 pages
http://dx.doi.org/10.1155/2014/419560
Research Article

Operator Ideal of Cesaro Type Sequence Spaces Involving Lacunary Sequence

1Department of Mathematics, Faculty of Science and Arts, King Abdulaziz University (KAU), P.O. Box 80200, Khulais 21589, Saudi Arabia
2Department of Mathematics, Faculty of Science, Ain Shams University, P.O. Box 1156, Abbassia, Cairo 11566, Egypt

Received 7 October 2013; Revised 13 January 2014; Accepted 15 January 2014; Published 26 February 2014

Academic Editor: Wenchang Sun

Copyright © 2014 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.

Abstract

The aim of this paper is to give the sufficient conditions on the sequence space defined in Lim (1977) such that the class of all bounded linear operators between any arbitrary Banach spaces with nth approximation numbers of the bounded linear operators in form an operator ideal.

1. Introduction

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 [1], Pietsch studied the operator ideals generated by the approximation numbers and classical sequence space . In [2], Faried and bakery [3] have studied the ideal of all bounded linear operators between any arbitrary Banach spaces whose sequence of approximation numbers belonged to the generalized Cesáro sequence space and Orlicz sequence space , when , ; these results coincide with that known for the classical sequence space . Bakery [4] has studied the operator ideals generated by the approximation numbers and generalized de La Vallée Poussin’s mean defined by Şimşek et al. [5]; these results coincide with that known in [2] for the generalized Cesáro sequence space. By , we indicate the space of all bounded linear operators from a normed space into a normed space . The set of nonnegative integers is denoted by and the real numbers by . By , we denote the space of all real sequences. A map which assigns to every operator a unique sequence is called an -function and the number is called the th -numbers of if the following conditions are satisfied:(a), for all ,(b), for all ,(c), for all , , and , where and are normed spaces,(d), for all , ,(e), if , for all ,(f) where is the identity operator on the Euclidean space .As examples of -numbers, we mention that approximation numbers , Gelfand numbers , Kolmogorov numbers , and Tichomirov numbers are defined by(I),(II), where is a metric injection (a metric injection is a one to one operator with closed range and with norm equal to one) from the space into a higher space for a suitable index set ,(III),(IV).All these numbers satisfy the following condition:(g) for all .The operator ideal is a subclass of , where and are Banach spaces such that its components satisfy the following conditions:(i), where denotes the 1-dimensional Banach space, where ;(ii)if , then for any scalars ;(iii)if , , and , then ; see [1, 6, 7].By a lacunary sequence , where , we mean an increasing sequence of nonnegative integers with . The intervals determined by are denoted by . We write . The space of lacunary strongly convergent sequences was defined by Freedman and denoted by It is well known that there exists very close connection between the space of lacunary strongly convergent sequences and the space of strongly Cesaro summable sequences. This connection can be found in [810].

For a sequence of positive real numbers with , for all , the generalized Cesaro 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 have been studied in [1113].

Remarks. (1) If , then we obtain the sequences space studied in [12, 13].

(2) If and , for all , then we obtain the sequences space studied in [14].

The idea of the paper is the following. We proceed in the following way: given a scalar sequence space , a pair of Banach spaces and , the space of bounded operators , and the approximation -numbers , , and , we define the space . Then, we study the following two problems:

Problem A (a linear problem). When (for which ) is an operator ideal.

Problem B (topological problems). When the ideal of the finite range operators in the class of Banach spaces is dense in and completeness of the components of the ideal.

Throughout this paper, the sequence is a bounded sequence of positive real numbers with the following:(a1)the sequence of positive real numbers is increasing and bounded with and ,(a2)the sequence is a nondecreasing sequence of positive real numbers tending to , with .

Also, we define , where 1 appears at the th place for all .

Recently different classes of paranormed sequence spaces have been introduced and their different properties have been investigated by Et et al. [15], Tripathy and Dutta [16, 17], and Tripathy and Borgogain [18], and see also [1923].

The following well-known inequality will be used throughout the paper. For any bounded sequence of positive numbers , and for all . See [24].

2. Preliminary and Notation

Definition 1. A class of linear sequence spaces is called a special space of sequences (sss) having three properties:(1) is a linear space and for each ;(2)if , and for all , then ; “that is, E is solid;”(3)if , then , where denotes the integral part of .

Example 2. is a special space of sequences for .

Example 3. defined in [14] is a special space of sequences for .

Example 4. Let be an Orlicz function satisfying -condition; then is a special space of sequences.

Example 5. studied in [3] is a special space of sequences, if is an increasing sequence of positive real numbers, and .

Example 6. is a special space of sequences, if the following conditions are satisfied:(1)the sequence of positive real numbers is increasing and bounded with and ;(2)the sequence is a nondecreasing sequence of positive real numbers tending to , and with .

Definition 7. are Banach spaces}, where .
We state the following result without proof.

Theorem 8. is an operator ideal if is a special space of sequences (sss).
We study here the operator ideals generated by the approximation numbers and the sequence space which are involving Lacunary sequence.

3. Main Results

Theorem 9. is an operator ideal, if conditions (a1) and (a2) are satisfied.

Proof. (1-i) Let ; since , , then .
(1-ii) Let , ; then ; we get , from (1-i) and (1-ii), and is a linear space.
To prove that for each , since . So, we get Hence, .
(2) Let for each ; then , since . Thus, .
(3) Let ; then we have So, .
Hence, from Theorem 8, it follows that is an operator ideal.

Corollary 10. is an operator ideal if is an increasing sequence of positive real numbers, and .

Corollary 11. is an operator ideal if .

Theorem 12. The linear space is dense in if conditions (a1) and (a2) are satisfied.

Proof. First, we show that every finite mapping belongs to . Since for each and is a linear space, then for every finite mapping , that is, the sequence contains only finitely many numbers different from zero. Now, we prove that . On taking , we obtain , and since , let ; then there exists a natural number such that for some , where . Since is decreasing for each , we get then there exists and with and since is a bounded sequence of positive real numbers, so on considering also . Then, there exists a natural number , with and . Since , then , so we can take Since is an increasing sequence, by using (7), (8), (9), and (10), we acquire This completes the proof.

Definition 13. A subclass of the special space of sequences called premodular special space of sequences characterized for the existence of a function , closely connected with the notion of modular but without assumption of the convexity, which satisfies the following:(i) for all and , where is the zero element of ;(ii)there exists a constant such that for all values of and for any scalar ;(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 -dense in ;(vii)for any , there exists a constant such that .It is obvious from condition (ii) that is continuous at the zero element of . The function defines a metrizable topology in endowed with this topology which is denoted by .

Example 14. is a premodular special space of sequences for with .

Example 15. is a premodular special space of sequences for with .

Example 16. Let be an Orlicz function satisfying -condition; then is a pre-modular special space of sequences with .

Example 17. If is an increasing sequence of positive real numbers, and , then is a premodular special space of sequences for , with .

Example 18. If the following conditions are satisfied:(1)the sequence of positive real numbers is increasing and bounded with and ;(2)the sequence is a nondecreasing sequence of positive real numbers tending to , , and with ; then is a premodular special space of sequences.

Theorem 19. with is a premodular special space of sequences, if conditions (a1) and (a2) are contented.

Proof. (i) Clearly, and .
(ii) Since is bounded, then there exists a constant such that for all values of and for any scalar .
(iii) For some numbers , we have the inequality for all .
(iv) Let for all ; then .
(v) There exist some numbers ; by using (iv), we have the inequality .
(vi) It is clear that the set of all finite sequences is -dense in .
(vii) For any , there exists a constant such that .

Theorem 20. Let be a normed space, let be a Banach space, and let conditions (a1) and (a2) be satisfied; then is complete.

Proof. Let be a Cauchy sequence in . Since with is a premodular special space of sequences, then, by using condition (vii) and since , we have , then is also a Cauchy sequence in . Since the space is a Banach space, then there exists such that and since for all , is continuous at 0 and, using (iii), we have for some .
Hence, as such .

Corollary 21. Let be a normed space, let be a Banach space, and let be an increasing sequence of positive real numbers with and ; then is complete.

Corollary 22. Let be a normed space, let be a Banach space, let and be an increasing sequence of positive real numbers with ; then is complete.

Conflict of Interests

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

Acknowledgments

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.

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