Abstract
Keywords:
approximation numbers; operator ideal; generalized Cesáro sequence space; Orlicz 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 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 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 . Example of snumbers, we mention approximation number , Gelfand numbers , Kolmogorov numbers and Tichomirov numbers defined by: All of these numbers satisfy the following condition:
(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.
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 , , then . See [13].
An Orlicz function is a function which is continuous, nondecreasing 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 [1418] and [19].
For any bounded sequence of positive numbers , we have the following wellknown 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 ,
(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 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 premodular special space of sequences for , with .
Example 2.3 is a premodular 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 satisfyingcondition and there exists a constantsuch that.
Proof
(1i) Let , since M is nondecreasing, we get , then .
(1ii) , since M satisfies condition, we get , for that , then from (1i) and (1ii) 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
we get , from (1i) and (1ii) is a linear space.
To show that for each , since we have . Thus, we get
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
and
since . Then there exists a natural number , with and . Since , then , so we can take
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
Corollary 3.8LetXbe a normed space, Ya Banach space andMbe an Orlicz function such thatMsatisfiescondition. 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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