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Mappings of type generalized de La Vallée Poussin’s mean

Abstract

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.

1 Introduction

By L(X,Y) we denote the space of all bounded linear operators from anormed space X into a normed space Y. The set of nonnegative integersis denoted by N={0,1,2,} and the real numbers by . By ω wedenote the space of all real sequences. A map which assigns to every operatorTL(X,Y) a unique sequence ( s n ( T ) ) n = 0 is called an s-function and the number s n (T) is called the n th s-numbers ofT if the following conditions are satisfied:

  1. (a)

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

  2. (b)

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

  3. (c)

    s n (RST)R s n (S)T for all TL( X 0 ,X), SL(X,Y) and RL(Y, Y 0 ).

  4. (d)

    s n (λT)=|λ| s n (T) for all TL(X,Y), λR.

  5. (e)

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

  6. (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:

  1. (I)

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

  2. (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.

  3. (III)

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

  4. (IV)

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

All these numbers satisfy the following condition:

  1. (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:

  1. (i)

    I K U, where K denotes the 1-dimensional Banach space, where UL.

  2. (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 .

  3. (iii)

    If VL( X 0 ,X), TU(X,Y) and RL(Y, Y 0 ), then RTVU( 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 nN, 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)={xω: n = 0 ( 1 n + 1 k = 0 n | x k | ) p n <}. Also, some geometric properties ofCes(p) are studied in [46] and [7].

Let Λ=( λ n ) be a nondecreasing sequence of positive real numberstending 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 kN.

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 withlim sup p n < and liminf p n >1,

(b2) the sequence ( λ n ) is a nondecreasing sequence of positive real numberstending 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 i th place for alliN.

Different classes of paranormed sequence spaces have been introduced and their differentproperties have been investigated. See [1215] 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 nN. See [17].

2 Preliminary and notation

Definition 2.1 A class of linear sequence spaces E is called a specialspace of sequences (sss) having the following conditions:

  1. (1)

    E is a linear space and e n E for each nN.

  2. (2)

    If xω, yE and | x n || y n | for all nN, then xEi.e., E is solid’.

  3. (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 for0<p<.

Example 2.3 ces p is a special space of sequences for1<p<.

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

Theorem 2.5 U E app is an operator ideal if E is a specialspace of sequences (sss).

Proof See [18]. □

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

Proof (1-i) Let x,yV(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+yV(λ,p).

(1-ii) Let λR, xV(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 λxV(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 mN, 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).

  1. (2)

    Let | x n || y n | for each nN, 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 yV(λ,p). Thus xV(λ,p).

  2. (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 realnumbers, lim n sup p n <and lim n inf p n >1.

Corollary 3.3 U ces p app is an operator ideal if1<p<.

Theorem 3.4 The linear spaceF(X,Y)is dense in U V ( λ , p ) app (X,Y)if conditions (b1) and (b2) aresatisfied.

Proof First we prove that every finite mapping TF(X,Y) belongs to U V ( λ , p ) app (X,Y). Since e m V(λ,p) for each mN and V(λ,p) is a linear space, then for every finite mappingTF(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 numbers>0 such that ρ( ( α n ( T ) ) n = s )< ε 2 h + 2 δ c for some c1, where δ=max{1, n = s ( 1 λ n ) p n }. Since α n (T) is decreasing for each nN, 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)2s 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 cantake

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

also α n (T)=inf{TA:AL(X,Y) and rank(A)n}. Then there exists a natural numberN>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 ifthere exists a function ρ:E[0,[ satisfying the following conditions:

  1. (i)

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

  2. (ii)

    there exists a constant l1 such that ρ(λx)l|λ|ρ(x) for all values of xE and for any scalar λ,

  3. (iii)

    for some numbers k1, we have the inequality ρ(x+y)k(ρ(x)+ρ(y)) for all x,yE,

  4. (iv)

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

  5. (v)

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

  6. (vi)

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

  7. (vii)

    for any λ>0, there exists a constant ζ>0 such that ρ(λ,0,0,0,)ζλρ(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 withthis topology which is denoted by E ρ .

Example 3.6 p is a pre-modular special space of sequences for0<p<, with ρ(x)= n = 0 | x n | p .

Example 3.7 ces p is a pre-modular special space of sequences for1<p<, with ρ(x)= n = 0 ( 1 n + 1 k = 0 n | x n | ) p .

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

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

  1. (ii)

    Since ( p n ) is bounded, then there exists a constant l1 such that ρ(λx)l|λ|ρ(x) for all values of xE and for any scalar λ.

  2. (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,yV(λ,p).

  3. (iv)

    Let | x n || y n | for all nN, 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 .

  4. (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 )).

  5. (vi)

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

  6. (vii)

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

Theorem 3.9 Let X be a normed space, Y be aBanach space, and let conditions (b1) and (b2) besatisfied, 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, byusing 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 existsTL(X,Y) such that T m T m 0 and since ( α n ( T m ) ) n = 0 E for all mN, ρ is continuous at θ andusing (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 3.10 Let X be a normed space, Y bea Banach space and( p n )be an increasing sequence of positive real numberswithlim sup p n <andlim inf p n >1, then U ces ( p ) app (X,Y)is complete.

Corollary 3.11 Let X be a normed space, Y bea Banach space and( p n )be an increasing sequence of positive real numberswith1<p<, then U ces p app (X,Y)is complete.

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Acknowledgements

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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Bakery, A.A. Mappings of type generalized de La Vallée Poussin’s mean. J Inequal Appl 2013, 518 (2013). https://doi.org/10.1186/1029-242X-2013-518

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Keywords

  • approximation numbers
  • operator ideal
  • generalized de La Vallée Poussin’s mean sequence space