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Mappings of type Orlicz and generalized Cesáro sequence space

Abstract

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 M , when M(t)= t p , 0<p<; our results coincide with that known for the classical sequence space p .

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 natural numbers will denote by N={0,1,2,} and the real numbers by . By ω, we denote the space of all real sequences. A map which assigns to every operator TL(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 of T 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 + m ( 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 . Example of s-numbers, we mention approximation number α r (T), Gelfand numbers c r (T), Kolmogorov numbers d r (T) and Tichomirov numbers d n (T) defined by: All of these numbers satisfy the following condition:

  7. (I)

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

  8. (II)

    c r (T)= a r ( 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 one) from the space Y into a higher space (Λ) for suitable index set Λ.

  9. (III)

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

  10. (IV)

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

  11. (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,Y are Banach spaces} such that its components {U(X,Y);X,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), RL(Y, Y 0 ) then RTVU( X 0 , Y 0 ). See [13].

An Orlicz function is a function M:[0,[[0,[ which is continuous, non-decreasing and convex with M(0)=0 and M(x)>0 for x>0, and M(x) as x. See [4, 5].

If convexity of Orlicz function M is replaced by M(x+y)M(x)+M(y). 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 Δ 2 -condition for all values of u, if there exists a constant k>0, such that M(2u)kM(u) (u0). The Δ 2 -condition is equivalent to M(lu)klM(u) for all values of u and for l>1. Lindentrauss and Tzafriri [10] used the idea of Orlicz function to construct Orlicz sequence space

M = { x ω : n = 0 M ( | x n | ρ ) < ,  for some  ρ > 0 } ,

which is a Banach space with respect to the norm

x=inf { ρ > 0 : n = 0 M ( | x n | ρ ) 1 } .

For M(t)= t p , 1p< the space M coincides with the classical sequence space p . 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 M(λx)λM(x) for all λ with 0<λ<1.

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 n )= { 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 n ) 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 n )= { x ω : n = 0 ( 1 n + 1 k = 0 n | x k | ) p n < } .

Also, some geometric properties of Ces( p n ) are studied by Sanhan and Suantai [13].

Throughout this paper, the sequence ( p n ) is a bounded sequence of positive real numbers, we denote e i =(0,0,,1,0,0,) where 1 appears at i th place for all iN. 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 ( p k ), we have the following well-known inequality | a k + b k | p k 2 h 1 ( | a k | p k + | b k | p k ),h= sup n p n , and p k 1 for all kN. 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. (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 .

We call such space E ρ a pre modular special space of sequences if there exists a function ρ:E[o,[, satisfies the following conditions:

  1. (i)

    ρ(x)0 x E ρ and ρ(θ)=0, 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 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 E ρ .

Example 2.2 p is a pre-modular special space of sequences for 0<p<, with ρ(x)= n = 0 | x n | p .

Example 2.3 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 .

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 } .

3 Main results

Theorem 3.1 U E app is an operator ideal if E is a special space of sequences (sss).

Proof To prove U E app is an operator ideal:

  1. (i)

    let AF(X,Y) and rank(A)=m for all mN, since E is a linear space and e n E for each nN, then ( α n ( A ) ) n = 0 =( α 0 (A), α 1 (A),, α m 1 (A),0,0,0,)= i = 0 m 1 α i (A) e i E; for that A U E app (X,Y), which implies F(X,Y) U E app (X,Y).

  2. (ii)

    Let T 1 , T 2 U E app (X,Y) and λ 1 , λ 2 R then from Definition 2.1 condition (3) we get ( α [ n 2 ] ( T 1 ) ) n = 0 E and ( α [ n 2 ] ( T 2 ) ) n = 0 E, since n2[ n 2 ], α n (T) is a decreasing sequence and from the definition of approximation numbers we get

    α n ( λ 1 T 1 + λ 2 T 2 ) α 2 [ n 2 ] ( λ 1 T 1 + λ 2 T 2 ) α [ n 2 ] ( λ 1 T 1 ) + α [ n 2 ] ( λ 2 T 2 ) | λ 1 | α [ n 2 ] ( T 1 ) + | λ 2 | α [ n 2 ] ( T 2 ) for each  n N .

Since E is a linear space and from Definition 2.1 condition (2) we get ( α n ( λ 1 T 1 + λ 2 T 2 ) ) n = 0 E, hence λ 1 T 1 + λ 2 T 2 U E app (X,Y).

  1. (iii)

    If VL( X 0 ,X), T U E app (X,Y) and RL(Y, Y 0 ), then we get ( α n ( T ) ) n = 0 E and since α n (RTV)R α n (T)V, from Definition 2.1 conditions (1) and (2) we get ( α n ( R T V ) ) n = 0 E, then RTV U E app ( X 0 , Y 0 ).

 □

Theorem 3.2 U M app is an operator ideal, if M is an Orlicz function satisfying Δ 2 -condition and there exists a constant l1 such that M(x+y)l(M(x)+M(y)).

Proof

(1-i) Let x,y M , since M is non-decreasing, we get n = 0 M(| x n + y n |)l[ n = 0 M(| x n |)+ n = 0 M(| y n |)]<, then x+y M .

(1-ii) λR, x M since M satisfies Δ 2 -condition, we get n = 0 M(|λ x n |)|λ|l n = 0 M(| x n |)<, for that λx M , then from (1-i) and (1-ii) M is a linear space over the field of numbers. Also e n M for each nN since i = 0 M(| e n (i)|)=M(1)<.

  1. (2)

    Let | x n || y n | for each nN, ( y n ) n = 0 M , since M is none decreasing, then we get n = 0 M(| x n |) n = 0 M(| y n |)<, then ( x n ) n = 0 M .

  2. (3)

    Let ( x n ) n = 0 M , n = 0 M(| x [ n 2 ] |)2 n = 0 M(| x n |)<, then ( x [ n 2 ] ) n = 0 M . Hence, from Theorem 3.1, it follows that U M app is an operator ideal.

 □

Theorem 3.3 U ces ( p n ) 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.

Proof

(1-i) Let x,yces( p n ) since

then x+yces( p n ).

(1-ii) Let λR, xces( p n ), then

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

we get λxces( p n ), from (1-i) and (1-ii) ces( p n ) is a linear space.

To show that e m ces( p n ) for each mN, since lim n inf p n >1 we have n = 0 ( 1 n + 1 ) p n <. Thus, we get

ρ( e m )= n = m ( 1 n + 1 k = 0 n | e m ( k ) | ) p n = n = m ( 1 n + 1 ) p n <.

Hence e m ces( p n ).

  1. (2)

    Let | x n || y n | for each nN, then

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

since yces( p n ). Thus, xces( p n ).

  1. (3)

    Let ( x n )ces( p n ), then we have

Hence, ( x [ n 2 ] ) n = 0 ces( p n ). Hence, from Theorem 3.1 it follows that U ces ( p n ) app is an operator ideal.

 □

Theorem 3.4 Let M be an Orlicz function. Then the linear space F(X,Y) is dense in U M app (X,Y).

Proof Define ρ(x)= n = 0 M(| x n |) on M . First we prove that every finite mapping TF(X,Y) belongs to U M app (X,Y). Since e m M for each mN and M is a linear space then for every finite mapping TF(X,Y) the sequence ( α n ( T ) ) n = 0 contains only finitely many numbers different from zero. To prove that U M app (X,Y) F ( X , Y ) ¯ , let T U M app (X,Y), we get ( α n ( T ) ) n = 0 M , and since n = 0 M( α n (T))<, let ε]0,1] then there exists a natural number s>0 such that n = s M( α n (T))< ε 4 , since ρ is none decreasing and α n (T) is decreasing for each nN, we get

sM ( α 2 s ( T ) ) n = s + 1 2 s M ( α n ( T ) ) n = s M ( α n ( T ) ) < ε 4 ,

then there exists A F 2 s (X,Y), rank(A)2s with M(TA)< ε 4 s , and by using the conditions of M we get

d ( T , A ) = ρ ( α n ( T A ) ) n = 0 = n = 0 M ( α n ( T A ) ) = n = 0 3 s 1 M ( α n ( T A ) ) + n = 3 s M ( α n ( T A ) ) n = 0 3 s 1 M ( T A ) + n = 3 s M ( α n ( T A ) ) 3 s M ( T A ) + n = s M ( α n + 2 s ( T A ) ) 3 s M ( T A ) + n = s M ( α n ( T ) ) < ε .

 □

Corollary 3.5 If 0<p< and M(t)= t p , we get U p app (X,Y)= F ( X , Y ) ¯ . See [3].

Theorem 3.6 The linear space F(X,Y) is dense in U ces ( p n ) app (X,Y), if ( p n ) is an increasing sequence of positive real numbers with lim n sup p n < and lim n inf p n >1.

Proof First we prove that every finite mapping TF(X,Y) belongs to U ces ( p n ) app (X,Y). Since e m ces( p n ) for each mN and ces( p n ) is a linear space, then for every finite mapping TF(X,Y) i.e. the sequence ( α n ( T ) ) n = 0 contains only finitely many numbers different from zero. Now we prove that U ces ( p n ) app (X,Y) F ( X , Y ) ¯ . Since lim n inf p n >1, we have n = 0 ( 1 n + 1 ) p n <, let T U ces ( p n ) app (X,Y) we get ( α n ( T ) ) n = 0 ces( p n ), 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 + 3 δ c for some c1, where δ=max{1, n = s ( 1 n + 1 ) p n }, since α n (T) is decreasing for each nN, we get

n = s + 1 2 s ( 1 n + 1 k = 0 n α 2 s ( T ) ) p n n = s + 1 2 s ( 1 n + 1 k = 0 n α n ( T ) ) p n n = s ( 1 n + 1 k = 0 n α k ( T ) ) p n < ε 2 h + 3 δ c ,
(1)

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

rank ( A ) 2 s with  n = 2 s + 1 3 s ( 1 n + 1 k = 0 n T A ) p n n = s + 1 2 s ( 1 n + 1 k = 0 n T A ) p n < ε 2 h + 3 δ c ,
(2)

and

sup n = s ( k = 0 s T A ) p n < ε 2 2 h + 2 δ ,
(3)

since α n (T)=inf{TA:AL(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 n + 1 k = 0 n T A ) p n < ε 2 h + 3 δ c ,
(4)

since ( p n ) is an increasing sequence and 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 n + 1 k = 0 n α k ( T A ) ) p n + n = 3 s ( 1 n + 1 k = 0 n α k ( T A ) ) p n n = 0 3 s ( 1 n + 1 k = 0 n T A ) p n + n = s ( 1 n + 1 k = 0 n + 2 s α k ( T A ) ) p n + 2 s 3 n = 0 s ( 1 n + 1 k = 0 n T A ) p n + n = s ( 1 n + 1 k = 0 2 s 1 α k ( T A ) + 1 n + 1 k = 2 s n + 2 s α k ( T A ) ) p n 3 n = 0 s ( 1 n + 1 k = 0 n T A ) p n + 2 h 1 ( n = s ( 1 n + 1 k = 0 2 s 1 α k ( T A ) ) p n + n = s ( 1 n + 1 k = 2 s n + 2 s α k ( T A ) ) p n ) 3 n = 0 s ( 1 n + 1 k = 0 n T A ) p n + 2 h 1 ( n = s ( 1 n + 1 k = 0 2 s 1 T A ) p n + n = s ( 1 n + 1 k = 0 n α k + 2 s ( T A ) ) p n ) 3 n = 0 s ( 1 n + 1 k = 0 n T A ) p n + 2 2 h 1 ( sup n = s ( k = 0 s T A ) p n ) n = s ( 1 n + 1 ) p n + 2 h 1 n = s ( 1 n + 1 k = 0 n α k ( T ) ) p n < ε .

 □

Theorem 3.7 Let X be a normed space, Y a Banach space and E ρ be a pre modular special space of sequences (sss), then U E ρ app (X,Y) is complete.

Proof Let ( T m ) be a Cauchy sequence in U E ρ app (X,Y), then by using Definition 2.1 condition (vii) and since U E ρ 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 Cauchy sequence in L(X,Y). Since the space L(X,Y) is a Banach space, then there exists TL(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 and using Definition 2.1(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 E as such T U E ρ app (X,Y). □

Corollary 3.8 Let X be a normed space, Y a Banach space and M be an Orlicz function such that M satisfies Δ 2 -condition. Then M is continuous at θ=(0,0,0,) and U M app (X,Y) is complete.

Corollary 3.9 Let X be a normed space, Y a Banach space and ( p n ) be an increasing sequence of positive real numbers with lim n sup p n < and lim n inf p n >1, then U ces ( p n ) app (X,Y) is complete.

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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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Correspondence to Nashat F Mohamed.

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NFM gave the idea of the article. AAB carried out the proofs and its application. All authors read and approved the final manuscript.

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Mohamed, N.F., Bakery, A.A. Mappings of type Orlicz and generalized Cesáro sequence space. J Inequal Appl 2013, 186 (2013). https://doi.org/10.1186/1029-242X-2013-186

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Keywords

  • approximation numbers
  • operator ideal
  • generalized Cesáro sequence space
  • Orlicz sequence space