Research

# Mappings of type generalized de La Vallée Poussin’s mean

Author Affiliations

Department of Mathematics, Faculty of Science and Arts, King Abdulaziz University (KAU), P.O. Box 80200, Khulais, Code 21589, Saudi Arabia

Department of Mathematics, Faculty of Science, Ain Shams University, Abbassia, P.O. Box 1156, Cairo, 11566, Egypt

Journal of Inequalities and Applications 2013, 2013:518  doi:10.1186/1029-242X-2013-518

 Received: 21 April 2013 Accepted: 9 September 2013 Published: 9 November 2013

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### 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):768-779, 2010). Our results 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.

##### Keywords:
approximation numbers; operator ideal; generalized de La Vallée Poussin’s mean sequence space

### 1 Introduction

By L ( X , Y ) 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 N = { 0 , 1 , 2 , } and the real numbers by ℝ. By ω we denote the space of all real sequences. A map which assigns to every operator T L ( X , Y ) a unique sequence ( s n ( T ) ) n = 0 is called an s-function and the number s n ( T ) is called the nth s-numbers of T if the following conditions are satisfied:

(a) T = s 0 ( T ) s 1 ( T ) 0 for all T L ( X , Y ) .

(b) s n ( T 1 + T 2 ) s n ( T 1 ) + T 2 for all T 1 , T 2 L ( X , Y ) .

(c) s n ( R S T ) R s n ( S ) T for all T L ( X 0 , X ) , S L ( X , Y ) and R L ( Y , Y 0 ) .

(d) s n ( λ T ) = | λ | s n ( T ) for all T L ( X , Y ) , λ R .

(e) rank ( T ) n if s n ( T ) = 0 for all T L ( X , Y ) .

(f)

s r ( I n ) = { 1 for  r < n , 0 for  r n ,

where I n is the identity operator on the Euclidean space 2 n .

As examples of s-numbers, we mention approximation numbers α n ( T ) , Gelfand numbers c n ( T ) , Kolmogorov numbers d n ( T ) and Tichomirov numbers d n ( T ) defined by:

(I) α n ( T ) = inf { T A : A L ( X , Y )  and  rank ( A ) n } .

(II) c n ( T ) = a n ( J Y T ) , where J Y 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 ( Λ ) for a suitable index set Λ.

(III) d n ( T ) = inf dim Y n sup x 1 inf y Y T x y .

(IV) d n ( T ) = d n ( J Y T ) .

All these numbers satisfy the following condition:

(g) s n + m ( T 1 + T 2 ) s n ( T 1 ) + s m ( T 2 ) for all T 1 , T 2 L ( X , Y ) .

An operator ideal U is a subclass of L = { L ( X , Y ) ; X  and  Y  are Banach spaces } such that its components { U ( X , Y ) ; X  and  Y  are Banach spaces } satisfy the following conditions:

(i) I K U , where K denotes the 1-dimensional Banach space, where U L .

(ii) If T 1 , T 2 U ( X , Y ) , then λ 1 T 1 + λ 2 T 2 U ( X , Y ) for any scalars λ 1 , λ 2 .

(iii) If V L ( X 0 , X ) , T U ( X , Y ) and R L ( Y , Y 0 ) , then R T V U ( X 0 , Y 0 ) . See [1,2] and [3].

For a sequence p = ( p n ) of positive real numbers with p n 1 , for all n N , the generalized Cesáro sequence space is defined by

Ces ( p ) = { x = ( x k ) ω : ρ ( λ x ) <  for some  λ > 0 } , where  ρ ( x ) = n = 0 ( 1 n + 1 k = 0 n | x k | ) p n .

The space Ces ( p ) is a Banach space with the norm x = inf { λ > 0 : ρ ( x λ ) 1 } .

If p = ( p n ) is bounded, we can simply write Ces ( p ) = { x ω : n = 0 ( 1 n + 1 k = 0 n | x k | ) p n < } . Also, some geometric properties of Ces ( p ) are studied in [4-6] and [7].

Let Λ = ( λ n ) be a nondecreasing sequence of positive real numbers tending to infinity, and let λ 0 = 1 and λ n + 1 λ n + 1 .

De La Vallée Poussin’s means of a sequence x = ( x k ) are defined as follows:

t n ( x ) = 1 λ n j I n | x j | , where  I n = [ n λ n + 1 , n ] ,  for  k N .

The generalized de La Vallée Poussin’s mean sequence space was defined in [8].

V ( λ , p ) = { x ω : ρ ( λ x ) <  for some  λ > 0 } , where  ρ ( x ) = n = 0 ( 1 λ n k I n | x k | ) p n .

The space V ( λ , p ) is a Banach space with the norm

x = inf { λ > 0 : ρ ( x λ ) 1 } .

If p = ( p n ) is bounded, we can simply write

V ( λ , p ) = { x ω : n = 0 ( 1 λ n k I n | x k | ) p n < } .

Also, some geometric properties of V ( λ , p ) are studied in [9,10] and [11].

Throughout this paper, the sequence ( p n ) is a bounded sequence of positive real numbers with

(b1) the sequence ( p n ) of positive real numbers is increasing and bounded with lim sup p n < and lim inf p n > 1 ,

(b2) the sequence ( λ n ) is a nondecreasing sequence of positive real numbers tending to infinity, λ 0 = 1 and λ n + 1 λ n + 1 with n = 0 ( 1 λ n ) p n < .

Also we define e i = ( 0 , 0 , , 1 , 0 , 0 , ) , where 1 appears at the ith place for all i N .

Different classes of paranormed sequence spaces have been introduced and their different properties have been investigated. See [12-15] and [16].

For any bounded sequence of positive numbers ( p n ) , we have the following well-known inequality | a n + b n | p n 2 h 1 ( | a n | p n + | b n | p n ) , h = sup n p n , and p n 1 for all n N . 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 e n E for each n N .

(2) If x ω , y E and | x n | | y n | for all n N , then x E i.e., E is solid’.

(3) If ( x n ) n = 0 E , then ( x [ n 2 ] ) n = 0 = ( x 0 , x 0 , x 1 , x 1 , x 2 , x 2 , ) E , where [ n 2 ] denotes the integral part of n 2 .

Example 2.2 p is a special space of sequences for 0 < p < .

Example 2.3 ces p is a special space of sequences for 1 < p < .

Definition 2.4 U E app : = { U E app ( X , Y ) ; X , Y  are Banach spaces } , where U E app ( X , Y ) : = { T L ( X , Y ) : ( α n ( T ) ) n = 0 E } .

Theorem 2.5 U E app is 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 3.1 U V ( λ , p ) app is an operator ideal, if conditions (b1) and (b2) are satisfied.

Proof (1-i) Let x , y V ( λ , p ) since

n = 0 ( 1 λ n k I n | x k + y k | ) p n 2 h 1 ( n = 0 ( 1 λ n k I n | x k | ) p n + n = 0 ( 1 λ n k I n | y k | ) p n ) ,

h = sup n p n , then x + y V ( λ , p ) .

(1-ii) Let λ R , x V ( λ , p ) , then

n = 0 ( 1 λ n k I n | λ x k | ) p n sup n | λ | p n n = 0 ( 1 λ n k I n | x k | ) p n < ,

we get λ x V ( λ , p ) , from (1-i) and (1-ii), V ( λ , p ) is a linear space.

To show that e m V ( λ , p ) for each m N , since n = 0 ( 1 λ n ) p n < . Thus we get

ρ ( e m ) = n = m ( 1 λ n k I n | e m ( k ) | ) p n = n = m ( 1 λ n ) p n < .

Hence e m V ( λ , p ) .

(2) Let | x n | | y n | for each n N , then n = 0 ( 1 λ n k I n | x k | ) p n n = 0 ( 1 λ n k I n | y k | ) p n since y V ( λ , p ) . Thus x V ( λ , p ) .

(3) Let ( x n ) V ( λ , p ) , then we have

n = 0 ( 1 λ n k I n | x [ k 2 ] | ) p n = n = 0 ( 1 λ 2 n k I 2 n | x [ k 2 ] | ) p 2 n + n = 0 ( 1 λ 2 n + 1 k I 2 n + 1 | x [ k 2 ] | ) p 2 n + 1 = n = 0 ( 1 λ 2 n ( ( k I n 2 | x k | ) + | x n | ) ) p n + n = 0 ( 1 λ 2 n + 1 ( k I n 2 | x k | ) ) p n 2 h 1 ( n = 0 ( 1 λ n ( 2 k I n | x k | ) ) p n + n = 0 ( 1 λ n k I n | x k | ) p n ) + 2 h n = 0 ( 1 λ n k I n | x k | ) p n 2 h 1 ( 2 h + 1 ) n = 0 ( 1 λ n k I n | x k | ) p n + 2 h n = 0 ( 1 λ n k I n | x k | ) p n ( 2 2 h 1 + 2 h 1 + 2 h ) n = 0 ( 1 λ n k I n | x k | ) p n < .

Hence ( x [ n 2 ] ) n = 0 V ( λ , p ) . Hence from Theorem 2.5 it follows that U V ( λ , p ) app is an operator ideal. □

Corollary 3.2 U ces ( p ) app is an operator ideal if ( p n ) is an increasing sequence of positive real numbers, lim n sup p n < and lim n inf p n > 1 .

Corollary 3.3 U ces p app is an operator ideal if 1 < p < .

Theorem 3.4The linear space F ( X , Y ) is dense in U V ( λ , p ) app ( X , Y ) if conditions (b1) and (b2) are satisfied.

Proof First we prove that every finite mapping T F ( X , Y ) belongs to U V ( λ , p ) app ( X , Y ) . Since e m V ( λ , p ) for each m N and V ( λ , p ) is a linear space, then for every finite mapping T F ( X , Y ) , i.e., the sequence ( α n ( T ) ) n = 0 contains only finitely many numbers different from zero. Now we prove that U V ( λ , p ) app ( X , Y ) F ( X , Y ) ¯ . Since letting T U V ( λ , p ) app ( X , Y ) we get ( α n ( T ) ) n = 0 V ( λ , p ) , and since ρ ( ( α n ( T ) ) n = 0 ) < , let ε ] 0 , 1 [ , then there exists a natural number s > 0 such that ρ ( ( α n ( T ) ) n = s ) < ε 2 h + 2 δ c for some c 1 , where δ = max { 1 , n = s ( 1 λ n ) p n } . Since α n ( T ) is decreasing for each n N , we get

n = s + 1 2 s ( 1 λ n k I n α 2 s ( T ) ) p n n = s + 1 2 s ( 1 λ n k I n α n ( T ) ) p n n = s ( 1 λ n k I n α k ( T ) ) p n < ε 2 h + 2 δ c , (1)

then there exists A F 2 s ( X , Y ) , rank ( A ) 2 s with

n = 2 s + 1 3 s ( 1 λ n k I n T A ) p n n = s + 1 2 s ( 1 λ n k I n T A ) p n < ε 2 h + 2 δ c , (2)

and since ( p n ) is a bounded sequence of positive real numbers, so we can take

sup n = s ( k I s T A ) p n < ε 2 h δ , (3)

also α n ( T ) = inf { T A : A L ( X , Y )  and  rank ( A ) n } . Then there exists a natural number N > 0 , A N with rank ( A N ) N and T A N 2 α N ( T ) . Since α n ( T ) n 0 , then

T A N N 0 , so we can take  n = 0 s ( 1 λ n k I n T A ) p n < ε 2 h + 3 δ c . (4)

Since ( p n ) is an increasing sequence, by using (1), (2), (3) and (4), we get

d ( T , A ) = ρ ( α n ( T A ) ) n = 0 = n = 0 3 s 1 ( 1 λ n k I n α k ( T A ) ) p n + n = 3 s ( 1 λ n k I n α k ( T A ) ) p n n = 0 3 s ( 1 λ n k I n T A ) p n + n = s ( 1 λ n k I n + 2 s α k ( T A ) ) p n + 2 s 3 n = 0 s ( 1 λ n k I n T A ) p n + n = s ( 1 λ n k I 2 s 1 α k ( T A ) + 1 λ n k I n + 2 s I 2 s 1 α k ( T A ) ) p n 3 n = 0 s ( 1 λ n k I n T A ) p n + 2 h 1 ( n = s ( 1 λ n k I 2 s 1 α k ( T A ) ) p n + n = s ( 1 λ n k I n + 2 s I 2 s 1 α k ( T A ) ) p n ) 3 n = 0 s ( 1 λ n k I n T A ) p n + 2 h 1 ( n = s ( 1 λ n k I s T A ) p n + n = s ( 1 λ n k I n α k + 2 s ( T A ) ) p n ) 3 n = 0 s ( 1 λ n k = 0 n T A ) p n + 2 h 1 sup n = s ( k I s T A ) p n n = s ( 1 λ n ) p n + 2 h 1 n = s ( 1 λ n k I n α k ( T ) ) p n < ε .

□

Definition 3.5 A class of special space of sequences (sss) E ρ is called a pre-modular special space of sequences if there exists a function ρ : E [ 0 , [ satisfying the following conditions:

(i) ρ ( x ) 0 x E ρ and ρ ( x ) = 0 x = θ , where θ is the zero element of E,

(ii) there exists a constant l 1 such that ρ ( λ x ) l | λ | ρ ( x ) for all values of x E and for any scalar λ,

(iii) for some numbers k 1 , we have the inequality ρ ( x + y ) k ( ρ ( x ) + ρ ( y ) ) for all x , y E ,

(iv) if | x n | | y n | for all n N , then ρ ( ( x n ) ) ρ ( ( y n ) ) ,

(v) for some numbers k 0 1 , we have the inequality ρ ( ( x n ) ) ρ ( ( x [ n 2 ] ) ) k 0 ρ ( ( x n ) ) ,

(vi) for each x = ( x ( i ) ) i = 0 E , there exists s N such that ρ ( x ( i ) ) i = s < . This means the set of all finite sequences is ρ-dense in E,

(vii) for any λ > 0 , there exists a constant ζ > 0 such that ρ ( λ , 0 , 0 , 0 , ) ζ λ ρ ( 1 , 0 , 0 , 0 , ) .

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 E ρ .

Example 3.6 p is a pre-modular special space of sequences for 0 < p < , with ρ ( x ) = n = 0 | x n | p .

Example 3.7 ces p is a pre-modular special space of sequences for 1 < p < , with ρ ( x ) = n = 0 ( 1 n + 1 k = 0 n | x n | ) p .

Theorem 3.8 V ( λ , p ) with ρ ( x ) = n = 0 ( 1 λ n k I n | x n | ) p n is a pre-modular special space of sequences if conditions (b1) and (b2) are satisfied.

Proof (i) Clearly, ρ ( x ) 0 and ρ ( x ) = 0 x = θ .

(ii) Since ( p n ) is bounded, then there exists a constant l 1 such that ρ ( λ x ) l | λ | ρ ( x ) for all values of x E and for any scalar λ.

(iii) For some numbers k = max ( 1 , 2 h 1 ) 1 , we have the inequality ρ ( x + y ) k ( ρ ( x ) + ρ ( y ) ) for all x , y V ( λ , p ) .

(iv) Let | x n | | y n | for all n N , then n = 0 ( 1 λ n k I n | x n | ) p n n = 0 ( 1 λ n k I n | y n | ) p n .

(v) There exist some numbers k 0 = 2 h 1 ( 2 h + 1 ) + 2 h 1 ; by using (iv) we have the inequality ρ ( ( x n ) ) ρ ( ( x [ n 2 ] ) ) k 0 ρ ( ( x n ) ) .

(vi) It is clear that the set of all finite sequences is ρ-dense in V ( λ , p ) .

(vii) For any λ > 0 , there exists a constant 0 < ζ < λ p 0 1 such that ρ ( λ , 0 , 0 , 0 , ) ζ λ ρ ( 1 , 0 , 0 , 0 , ) . □

Theorem 3.9LetXbe a normed space, Ybe a Banach space, and let conditions (b1) and (b2) be satisfied, then U V ρ ( λ , p ) app ( X , Y ) is complete.

Proof Let ( T m ) be a Cauchy sequence in U V ρ ( λ , p ) app ( X , Y ) . Since V ( λ , p ) with ρ ( x ) = n = 0 ( 1 λ n k I n | x n | ) p n is a pre-modular special space of sequences, then, by using condition (vii) and since U V ρ ( λ , p ) app ( X , Y ) L ( X , Y ) , we have ρ ( ( α n ( T i T j ) ) n = 0 ) ρ ( α 0 ( T i T j ) , 0 , 0 , 0 , ) = ρ ( T i T j , 0 , 0 , 0 , ) ζ T i T j ρ ( 1 , 0 , 0 , 0 , ) , then ( T m ) is also a Cauchy sequence in L ( X , Y ) . Since the space L ( X , Y ) is a Banach space, then there exists T L ( X , Y ) such that T m T m 0 and since ( α n ( T m ) ) n = 0 E for all m N , ρ is continuous at θ and using (iii), we have

ρ ( α n ( T ) ) n = 0 = ρ ( α n ( T T m + T m ) ) n = 0 k ρ ( α [ n 2 ] ( T m T ) ) n = 0 + k ρ ( α [ n 2 ] ( T m ) ) n = 0 k ρ ( ( T m T ) n = 0 ) + k ρ ( α n ( T m ) ) n = 0 < ε for some  k 1 .

Hence ( α n ( T ) ) n = 0 V ρ ( λ , p ) as such T U V ρ ( λ , p ) app ( X , Y ) . □

Corollary 3.10LetXbe a normed space, Ybe a Banach space and ( p n ) be an increasing sequence of positive real numbers with lim sup p n < and lim inf p n > 1 , then U ces ( p ) app ( X , Y ) is complete.

Corollary 3.11LetXbe a normed space, Ybe a Banach space and ( p n ) be an increasing sequence of positive real numbers with 1 < p < , then U ces p app ( X , Y ) is 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.

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