Mappings of type generalized de La Vallée Poussin’s mean
© Bakery; licensee Springer. 2013
Received: 21 April 2013
Accepted: 9 September 2013
Published: 9 November 2013
In the present paper, we study the operator ideals generated by the approximationnumbers and generalized de La Vallée Poussin’s mean defined in(Şimşek et al. in J. Comput. Anal. Appl. 12(4):768-779, 2010). Ourresults coincide with those in (Faried and Bakery in J. Inequal. Appl. 2013,doi:10.1186/1029-242X-2013-186) for the generalized Cesáro sequence space.
Keywordsapproximation numbers operator ideal generalized de La Vallée Poussin’s mean sequence space
for all .
for all .
for all , and .
for all , .
if for all .
where is the identity operator on the Euclidean space.
, 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 Y into a higher space for a suitable index set Λ.
for all .
, where K denotes the 1-dimensional Banach space, where .
If , then for any scalars , .
The space is a Banach space with the norm .
Let be a nondecreasing sequence of positive real numberstending to infinity, and let and .
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 numberstending to infinity, and with .
Also we define , where 1 appears at the i th place for all.
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 .
Example 2.2 is a special space of sequences for.
Example 2.3 is a special space of sequences for.
Definition 2.4, where .
Theorem 2.5is an operator ideal if E is a specialspace of sequences (sss).
Proof See . □
We give here the sufficient conditions on the generalized de La ValléePoussin’s mean such that the class of all bounded linear operators between anyarbitrary Banach spaces with n th approximation numbers of the bounded linearoperators in the generalized de La Vallée Poussin’s mean form an operatorideal.
3 Main results
Theorem 3.1is an operator ideal, if conditions (b1)and (b2) are satisfied.
, then .
we get , from (1-i) and (1-ii), is a linear space.
Let for each , then since . Thus .
- (3)Let , then we have
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 realnumbers, and.
Corollary 3.3is an operator ideal if.
Theorem 3.4 The linear spaceis dense inif conditions (b1) and (b2) aresatisfied.
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 from condition (ii) that ρ is continuous at θ.The function ρ defines a metrizable topology in E endowed withthis topology which is denoted by .
Example 3.6 is a pre-modular special space of sequences for, with .
Example 3.7 is a pre-modular special space of sequences for, with .
Theorem 3.8withis a pre-modular special space of sequences ifconditions (b1) and (b2) are satisfied.
Since is bounded, then there exists a constant such that for all values of and for any scalar λ.
For some numbers , we have the inequality for all .
Let for all , then .
There exist some numbers ; by using (iv) we have the inequality .
It is clear that the set of all finite sequences is ρ-dense in .
For any , there exists a constant such that . □
Theorem 3.9 Let X be a normed space, Y be aBanach space, and let conditions (b1) and (b2) besatisfied, thenis complete.
Hence as such . □
Corollary 3.10 Let X be a normed space, Y bea Banach space andbe an increasing sequence of positive real numberswithand, thenis complete.
Corollary 3.11 Let X be a normed space, Y bea Banach space andbe an increasing sequence of positive real numberswith, thenis complete.
The author is most grateful to the editor and anonymous referee for careful readingof the paper and valuable suggestions which helped in improving an earlier version ofthis paper.
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