Abstract
In the present paper, we study the operator ideals generated by the approximation numbers and generalized de La Vallée Poussin’s mean defined in (Şimşek et al. in J. Comput. Anal. Appl. 12(4):768779, 2010). Our results coincide with those in (Faried and Bakery in J. Inequal. Appl. 2013, doi:10.1186/1029242X2013186) for the generalized Cesáro sequence space.
Keywords:
approximation numbers; operator ideal; generalized de La Vallée Poussin’s mean sequence space1 Introduction
By we denote the space of all bounded linear operators from a normed space X into a normed space Y. The set of nonnegative integers is denoted 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 sfunction and the number is called the nth snumbers of T if the following conditions are satisfied:
(f)
where is the identity operator on the Euclidean space .
As examples of snumbers, we mention approximation numbers , Gelfand numbers , Kolmogorov numbers and Tichomirov numbers defined by:
(II) , where is a metric injection (a metric injection is a onetoone operator with closed range and with norm equal to one) from the space Y into a higher space for a suitable index set Λ.
All these numbers satisfy the following condition:
An operator ideal U is a subclass of such that its components satisfy the following conditions:
(i) , where K denotes the 1dimensional Banach space, where .
(ii) If , then for any scalars , .
(iii) If , and , then . See [1,2] and [3].
For a sequence of positive real numbers with , for all , the generalized Cesáro sequence space is defined by
The space is a Banach space with the norm .
If is bounded, we can simply write . Also, some geometric properties of are studied in [46] and [7].
Let be a nondecreasing sequence of positive real numbers tending to infinity, and let and .
De La Vallée Poussin’s means of a sequence are defined as follows:
The generalized de La Vallée Poussin’s mean sequence space was defined in [8].
The space is a Banach space with the norm
If is bounded, we can simply write
Also, some geometric properties of are studied in [9,10] and [11].
Throughout this paper, the sequence is a bounded sequence of positive real numbers with
(b1) the sequence of positive real numbers is increasing and bounded with and ,
(b2) the sequence is a nondecreasing sequence of positive real numbers tending to infinity, and with .
Also we define , where 1 appears at the ith place for all .
Different classes of paranormed sequence spaces have been introduced and their different properties have been investigated. See [1215] and [16].
For any bounded sequence of positive numbers , we have the following wellknown inequality , , and for all . See [17].
2 Preliminary and notation
Definition 2.1 A class of linear sequence spaces E is 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 .
Example 2.2 is a special space of sequences for .
Example 2.3 is a special space of sequences for .
Theorem 2.5is an operator ideal ifEis a special space of sequences (sss).
Proof See [18]. □
We give here the sufficient conditions on the generalized de La Vallée Poussin’s mean such that the class of all bounded linear operators between any arbitrary Banach spaces with nth approximation numbers of the bounded linear operators in the generalized de La Vallée Poussin’s mean form an operator ideal.
3 Main results
Theorem 3.1is an operator ideal, if conditions (b1) and (b2) are satisfied.
we get , from (1i) and (1ii), is a linear space.
To show that for each , since . Thus we get
(2) Let for each , then since . Thus .
Hence . Hence from Theorem 2.5 it follows that is an operator ideal. □
Corollary 3.2is an operator ideal ifis an increasing sequence of positive real numbers, and.
Corollary 3.3is an operator ideal if.
Theorem 3.4The linear spaceis dense inif conditions (b1) and (b2) are satisfied.
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 letting we get , and since , let , then there exists a natural number such that for some , where . Since is decreasing for each , we get
and since is a bounded sequence of positive real numbers, so we can take
also . Then there exists a natural number , with and . Since , then
Since is an increasing sequence, by using (1), (2), (3) and (4), we get
□
Definition 3.5 A class of special space of sequences (sss) is called a premodular special space of sequences if there exists a function satisfying 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 ,
(v) for some numbers , we have the inequality ,
(vi) for each , there exists such that . This means the set of all finite sequences is ρdense in E,
(vii) for any , there exists a constant such that .
It is clear from condition (ii) that ρ is continuous at θ. The function ρ defines a metrizable topology in E endowed with this topology which is denoted by .
Example 3.6 is a premodular special space of sequences for , with .
Example 3.7 is a premodular special space of sequences for , with .
Theorem 3.8withis a premodular special space of sequences if conditions (b1) and (b2) are satisfied.
(ii) Since is bounded, then 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 .
(v) There exist some numbers ; by using (iv) we have the inequality .
(vi) It is clear that the set of all finite sequences is ρdense in .
(vii) For any , there exists a constant such that . □
Theorem 3.9LetXbe a normed space, Ybe a Banach space, and let conditions (b1) and (b2) be satisfied, thenis 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 , ρ is continuous at θ and using (iii), we have
Corollary 3.10LetXbe a normed space, Ybe a Banach space andbe an increasing sequence of positive real numbers withand, thenis complete.
Corollary 3.11LetXbe a normed space, Ybe a Banach space andbe an increasing sequence of positive real numbers with, thenis complete.
Competing interests
The author declares that he has no competing interests.
Acknowledgements
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 this paper.
References

Kalton, NJ: Spaces of compact operators. Math. Ann.. 208, 267–278 (1974). Publisher Full Text

Lima, Å, Oja, E: Ideals of finite rank operators, intersection properties of balls, and the approximation property. Stud. Math.. 133, 175–186 (1999)

Pietsch, A: Operator Ideals, NorthHolland, Amsterdam (1980)

Sanhan, W, Suantai, S: On knearly uniformly convex property in generalized Cesáro sequence space. Int. J. Math. Math. Sci.. 57, 3599–3607 (2003)

Savas, E, Karakaya, V, Şimşek, N: Some type new sequence spaces and their geometric properties. Abstr. Appl. Anal.. 2009, (2009) Article ID 69697

Şimşek, N, Karakaya, V: On some geometrical properties of generalized modular spaces of Cesáro type defined by weighted means. J. Inequal. Appl.. 2009, (2009) Article ID 932734. 40G05 (26E60)

Karakaya, V: Some geometric properties of sequence spaces involving lacunary sequence. J. Inequal. Appl.. 2007, (2007) Article ID 81028

Şimşek, N, Savas, E, Karakaya, V: Some geometric and topological properties of a new sequence space defined by de La Vallée Poussin mean. J. Comput. Anal. Appl.. 12(4), 768–779 (2010)

Şimşek, N, Savas, E, Karakaya, V: On geometrical properties of some Banach spaces. Appl. Math. Inf. Sci.. 7(1), 295–300 (2013). Publisher Full Text

Şimşek, N: On some geometric properties of sequence space defined by de La Vallée Poussin mean. J. Comput. Anal. Appl.. 13(3), 565–573 (2011)

Çinar, M, Karakaş, M, Et, M: Some geometric properties of the metric space . J. Inequal. Appl.. 2013, (2013) Article ID 28

Rath, D, Tripathy, BC: Matrix maps on sequence spaces associated with sets of integers. Indian J. Pure Appl. Math.. 27(2), 197–206 (1996)

Tripathy, BC, Sen, M: On generalized statistically convergent sequences. Indian J. Pure Appl. Math.. 32(11), 1689–1694 (2001)

Tripathy, BC, Chandra, P: On some generalized difference paranormed sequence spaces associated with multiplier sequences defined by modulus function. Anal. Theory Appl.. 27(1), 21–27 (2011). Publisher Full Text

Tripathy, BC: Matrix transformations between some classes of sequences. J. Math. Anal. Appl.. 206, 448–450 (1997). Publisher Full Text

Tripathy, BC: On generalized difference paranormed statistically convergent sequences. Indian J. Pure Appl. Math.. 35(5), 655–663 (2004)

Altay, B, Başar, F: Generalization of the sequence space derived by weighted means. J. Math. Anal. Appl.. 330(1), 147–185 (2007)

Faried, N, Bakery, AA: Mappings of type Orlicz and generalized Cesáro sequence space. J. Inequal. Appl. (2013). BioMed Central Full Text