Mappings of type Orlicz and generalized Cesáro sequence space
© Mohamed and Bakery; licensee Springer 2013
Received: 7 January 2013
Accepted: 5 April 2013
Published: 18 April 2013
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 .
Keywordsapproximation numbers operator ideal generalized Cesáro sequence space Orlicz sequence space
, for all .
, for all .
, for all , and .
, for all , .
If , for all .
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:
, 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.
for all .
, where K denotes the 1-dimensional Banach space, where .
If , then for any scalars , .
Remark 1.1 Let M be an Orlicz function then for all λ with .
Also, some geometric properties of are studied by Sanhan and Suantai .
Throughout this paper, the sequence is a bounded sequence of positive real numbers, we denote where 1 appears at i th place for all . Different classes of paranormed sequence spaces have been introduced and their different properties have been investigated. See [14–18] and .
For any bounded sequence of positive numbers , we have the following well-known inequality , and for all . See .
2 Preliminary and notation
E is a linear space and , for each .
If , and , for all , then ‘i.e. E is solid’,
if , then , where denotes the integral part of .
and , where θ is the zero element of E,
there exists a constant such that for all values of and for any scalar λ,
for some numbers , we have the inequality , for all ,
if , for all then ,
for some numbers we have the inequality ,
for each there exists such that . This means the set of all finite sequences is ρ-dense in E.
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 .
3 Main results
Theorem 3.1 is an operator ideal if E is a special space of sequences (sss).
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
If , and , then we get and since , from Definition 2.1 conditions (1) and (2) we get , then .
Theorem 3.2 is an operator ideal, if M is an Orlicz function satisfying -condition and there exists a constant such that .
(1-i) Let , since M is non-decreasing, we get , then .
Let for each , , since M is none decreasing, then we get , then .
Let , , then . Hence, from Theorem 3.1, it follows that is an operator ideal.
Theorem 3.3 is an operator ideal, if is an increasing sequence of positive real numbers, and .
we get , from (1-i) and (1-ii) is a linear space.
- (2)Let for each , then
Hence, . Hence, from Theorem 3.1 it follows that is an operator ideal.
Theorem 3.4 Let M be an Orlicz function. Then the linear space is dense in .
Corollary 3.5 If and , we get . See .
Theorem 3.6 The linear space is dense in , if is an increasing sequence of positive real numbers with and .
Theorem 3.7 Let X be a normed space, Y a Banach space and be a pre modular special space of sequences (sss), then is complete.
Hence as such . □
Corollary 3.8 Let X be a normed space, Y a Banach space and M be an Orlicz function such that M satisfies -condition. Then M is continuous at and is complete.
Corollary 3.9 Let X be a normed space, Y a Banach space and be an increasing sequence of positive real numbers with and , then is complete.
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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