R E S E A R C H
Open Access
Mappings of type Orlicz and generalized
Cesáro sequence space
Nashat F Mohamed
1*and Awad A Bakery
1,2*Correspondence: [email protected] 1Department of Mathematics, Faculty of Science, Ain Shams University, Cairo, Egypt Full list of author information is available at the end of the article
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, whenM(t) =tp, 0 <p<∞; our results
coincide with that known for the classical sequence space
p.
Keywords: approximation numbers; operator ideal; generalized Cesáro sequence space; Orlicz sequence space
1 Introduction
ByL(X,Y), we denote the space of all bounded linear operators from a normed space X into a normed spaceY. The set of natural numbers will denote byN={, , , . . .}and the real numbers byR. Byω, we denote the space of all real sequences. A map which assigns to every operatorT∈L(X,Y) a unique sequence (sn(T))∞n=is called ans-function and the
numbersn(T) is called thenths-numbers ofTif the following conditions are satisfied:
(a) T=s(T)≥s(T)≥ · · · ≥, for allT∈L(X,Y).
(b) sn+m(T+T)≤sn(T) +T, for allT,T∈L(X,Y).
(c) sn(RST)≤ Rsn(S)T, for allT∈L(X,X),S∈L(X,Y)andR∈L(Y,Y).
(d) sn(λT) =|λ|sn(T), for allT∈L(X,Y),λ∈R.
(e) rank(T)≤nIfsn(T) = , for allT∈L(X,Y).
(f ) sr(In) =
forr<n,
forr≥n, whereInis the identity operator on the Euclidean space
n
.
Example ofs-numbers, we mention approximation numberαr(T), Gelfand numbers
cr(T), Kolmogorov numbersdr(T)and Tichomirov numbersd∗n(T)defined by:
(I) αr(T) =inf{T–A:A∈L(X,Y)andrank(A)≤r}.
(II) cr(T) =ar(JYT), whereJY is a metric injection (a metric injection is a one to
one operator with closed range and with norm equal one) from the spaceY
into a higher space∞()for suitable index set. (III) dn(T) =infdimY≤nsupx≤infy∈YTx–y.
(IV) d∗r(T) =dr(JYT).
All of these numbers satisfy the following condition: (g) sn+m(T+T)≤sn(T) +sm(T)for allT,T∈L(X,Y).
An operator idealUis a subclass ofL={L(X,Y);X,Yare Banach spaces}such that its components{U(X,Y);X,Yare Banach spaces}satisfy the following conditions:
(i) IK∈U, whereKdenotes the -dimensional Banach space, whereU⊂L.
(ii) IfT,T∈U(X,Y), thenλT+λT∈U(X,Y)for any scalarsλ,λ.
(iii) IfV∈L(X,X),T∈U(X,Y),R∈L(Y,Y)thenRTV∈U(X,Y). See [–].
An Orlicz function is a function M: [,∞[→ [,∞[ which is continuous, non-decreasing and convex withM() = andM(x) > forx> , andM(x)→ ∞asx→ ∞. See [, ].
If convexity of Orlicz functionMis replaced byM(x+y)≤M(x) +M(y). Then this func-tion is called modulus funcfunc-tion, introduced by Nakano []; also, see [, ] and []. An Or-licz functionMis said to satisfy-condition for all values of u, if there exists a constant
k> , such thatM(u)≤kM(u) (u≥). The-condition is equivalent toM(lu)≤klM(u)
for all values of u and forl> . Lindentrauss and Tzafriri [] used the idea of Orlicz func-tion to construct Orlicz sequence space
M=
x∈ω: ∞
n=
M |
xn|
ρ
<∞, for someρ>
,
which is a Banach space with respect to the norm
x=inf
ρ> : ∞
n=
M
|xn|
ρ
≤
.
For M(t) =tp, ≤p<∞ the space
M coincides with the classical sequence spacep. Recently, different classes of sequences have been introduced by using an Orlicz function. See [] and [].
Remark . LetMbe an Orlicz function thenM(λx)≤λM(x) for allλwith <λ< .
For a sequencep= (pn) of positive real numbers withpn≥, for alln∈Nthe generalized Cesáro sequence space is defined by
Ces(pn) =
x= (xk)∈ω:ρ(λx) <∞for someλ> ,
where
ρ(x) = ∞
n=
n+
n
k=
|xk| pn
.
The spaceCes(pn) is a Banach space with the norm
x=inf
λ> :ρ
x
λ
≤
.
Ifp= (pn) is bounded, we can simply write
Ces(pn) =
x∈ω: ∞
n=
n+
n
k=
|xk| pn
<∞
.
For any bounded sequence of positive numbers (pk), we have the following well-known inequality|ak+bk|pk≤h–(|ak|pk+|bk|pk),h=supnpn, andpk≥ for allk∈N. See []. 2 Preliminary and notation
Definition . A class of linear sequence spacesE, called a special space of sequences (sss) having the following conditions:
() Eis a linear space anden∈E, for eachn∈N.
() Ifx∈ω,y∈Eand|xn| ≤ |yn|, for alln∈N, thenx∈E‘i.e.Eis solid’,
() if(xn)∞n=∈E, then(x[n])∞n== (x,x,x,x,x,x, . . .)∈E, where [n] denotes the
integral part ofn.
We call such spaceEρa pre modular special space of sequences if there exists a function ρ:E→[o,∞[, satisfies the following conditions:
(i) ρ(x)≥∀x∈Eρandρ(θ) = , whereθ is the zero element ofE,
(ii) there exists a constantl≥such thatρ(λx)≤l|λ|ρ(x)for all values ofx∈Eand for any scalarλ,
(iii) for some numbersk≥, we have the inequalityρ(x+y)≤k(ρ(x) +ρ(y)), for all
x,y∈E,
(iv) if|xn| ≤ |yn|, for alln∈Nthenρ((xn))≤ρ((yn)),
(v) for some numbersk≥we have the inequalityρ((xn))≤ρ((x[n]))≤kρ((xn)),
(vi) for eachx= (x(i))∞i=∈Ethere existss∈Nsuch thatρ(x(i))∞i=s<∞. This means the set of all finite sequences isρ-dense inE.
(vii) for anyλ> there exists a constantζ> such that
ρ(λ, , , , . . .)≥ζ λρ(, , , , . . .).
It is clear that from condition (ii) thatρis continuous atθ. The functionρdefines a metriz-able topology inEendowed with this topology is denoted byEρ.
Example . p is a pre-modular special space of sequences for <p<∞, withρ(x) =
∞
n=|xn|p.
Example . cespis a pre-modular special space of sequences for <p<∞, withρ(x) =
∞
n=(n+
n
k=|xn|)p.
Definition .
UEapp:=UEapp(X,Y);X,Yare Banach spaces ,
where
UEapp(X,Y) :=T∈L(X,Y) :αn(T) ∞
n=∈E .
3 Main results
Theorem . UEappis an operator ideal if E is a special space of sequences(sss).
Proof To proveUEappis an operator ideal:
(i) letA∈F(X,Y)andrank(A) =mfor allm∈N, since E is a linear space anden∈E
for eachn∈N, then(αn(A))∞n== (α(A),α(A), . . . ,αm–(A), , , , . . .) =
m–
i= αi(A)ei∈E; for thatA∈UEapp(X,Y), which impliesF(X,Y)⊂U
app
(ii) LetT,T∈UEapp(X,Y)andλ,λ∈Rthen from Definition . condition () we get
(α[n
](T))
∞
n=∈Eand(α[n](T))
∞
n=∈E, sincen≥[n],αn(T)is a decreasing
sequence and from the definition of approximation numbers we get
αn(λT+λT)≤α[n](λT+λT)≤α[n](λT) +α[n](λT)
≤ |λ|α[n](T) +|λ|α[n](T) for eachn∈N.
SinceEis a linear space and from Definition . condition () we get
(αn(λT+λT))∞n=∈E, henceλT+λT∈UEapp(X,Y).
(iii) IfV∈L(X,X),T∈UEapp(X,Y)andR∈L(Y,Y), then we get(αn(T))∞n=∈Eand
sinceαn(RTV)≤ Rαn(T)V, from Definition . conditions () and () we get (αn(RTV))∞n=∈E, thenRTV∈U
app
E (X,Y).
Theorem . Uapp
M is an operator ideal,if M is an Orlicz function satisfying-condition and there exists a constant l≥such that M(x+y)≤l(M(x) +M(y)).
Proof
(-i) Let x,y∈M, sinceMis non-decreasing, we get ∞
n=M(|xn+yn|)≤l[
∞
n=M(|xn|) +
∞
n=M(|yn|)] <∞, thenx+y∈M.
(-ii) λ∈R,x∈MsinceMsatisfies-condition, we get
∞
n=M(|λxn|)≤ |λ|l
∞
n=M(|xn|) <∞, for thatλx∈M, then from (-i) and
(-ii)Mis a linear space over the field of numbers. Alsoen∈Mfor eachn∈N
since∞i=M(|en(i)|) =M() <∞.
() Let|xn| ≤ |yn|for eachn∈N, (yn)∞n=∈M, sinceMis none decreasing, then we
get∞n=M(|xn|)≤∞n=M(|yn|) <∞, then(xn)∞n=∈M.
() Let(xn)∞n=∈M,
∞
n=M(|x[n]|)≤
∞
n=M(|xn|) <∞, then(x[n])
∞ n=∈M.
Hence, from Theorem ., it follows thatUappM is an operator ideal.
Theorem . Ucesapp(pn)is an operator ideal,if(pn)is an increasing sequence of positive real
numbers,limn→∞suppn<∞andlimn→∞infpn> .
Proof
(-i) Letx,y∈ces(pn)since
∞
n=
n+
n
k=
|xk+yk| pn
≤h–
∞
n=
n+
n
k=
|xk| pn
+ ∞
n=
n+
n
k=
|yk| pn
,
h=sup
n
pn,
thenx+y∈ces(pn).
(-ii) Letλ∈R,x∈ces(pn), then ∞
n=
n+
n
k=
|λxk| pn
≤sup
n |
λ|pn ∞
n=
n+
n
k=
|xk| pn
<∞,
To show thatem∈ces(pn) for eachm∈N, sincelimn→∞infpn> we have
∞
n=(n+)pn<
∞. Thus, we get
ρ(em) = ∞
n=m
n+
n
k=
em(k) pn
= ∞
n=m
n+
pn <∞.
Henceem∈ces(pn).
() Let|xn| ≤ |yn|for eachn∈N, then
∞
n=
n+
n
k=
|λxk| pn
≤sup
n |
λ|pn ∞
n=
n+
n
k=
|yk| pn
<∞,
sincey∈ces(pn). Thus,x∈ces(pn).
() Let(xn)∈ces(pn), then we have
∞
n=
n+
n
k=
|x[k
]| pn = ∞ n= n+
n
k=
|x[k
]|
pn + ∞ n= n+
n+
k=
|x[k
]|
pn+
= ∞ n= n+
n
k=
|xk|
+|xn|
pn + ∞ n= n+
n
k=
|xk|
pn
≤h–
∞
n=
n+
n
k=
|xk|
pn + ∞ n= n+
n
k=
|xk| pn + ∞ n= n+
n
k=
|xk| pn
≤h–h+ ∞
n=
n+
n
k=
|xk| pn + ∞ n= n+
n
k=
|xk| pn
≤h–+ h–+ ∞
n=
n+
n
k=
|xk| pn
<∞.
Hence,(x[n])∞n=∈ces(pn). Hence, from Theorem . it follows thatUcesapp(pn)is an
operator ideal.
Theorem . Let M be an Orlicz function. Then the linear space F(X,Y) is dense in UappM(X,Y).
Proof Defineρ(x) =∞n=M(|xn|) onM. First we prove that every finite mappingT ∈
F(X,Y) belongs toUappM(X,Y). Sinceem∈Mfor eachm∈NandMis a linear space then for every finite mappingT∈F(X,Y) the sequence (αn(T))∞n=contains only finitely many
numbers different from zero. To prove thatUappM(X,Y)⊆F(X,Y), letT∈U
app
M(X,Y), we get (αn(T))∞n=∈M, and since
∞
decreasing for eachn∈N, we get
sMαs(T)
≤
s
n=s+
Mαn(T)
≤ ∞
n=s
Mαn(T)
< ε ,
then there existsA∈Fs(X,Y),rank(A)≤swithM(T–A) <εs, and by using the con-ditions ofMwe get
d(T,A) =ραn(T–A) ∞
n==
∞
n=
Mαn(T–A)
=
s–
n=
Mαn(T–A)
+ ∞
n=s
Mαn(T–A)
≤
s–
n=
MT–A+ ∞
n=s
Mαn(T–A)
≤sMT–A+ ∞
n=s
Mαn+s(T–A)
≤sMT–A+ ∞
n=s
Mαn(T)
<ε.
Corollary . If <p<∞andM(t) =tp,we get Uappp (X,Y) =F(X,Y).See[].
Theorem . The linear space F(X,Y)is dense in Ucesapp(pn)(X,Y),if(pn)is an increasing
sequence of positive real numbers withlimn→∞suppn<∞andlimn→∞infpn> .
Proof First we prove that every finite mappingT∈F(X,Y) belongs toUcesapp(pn)(X,Y). Since
em ∈ces(pn) for each m∈N andces(pn) is a linear space, then for every finite map-pingT ∈F(X,Y)i.e. the sequence (αn(T))∞n= contains only finitely many numbers
dif-ferent from zero. Now we prove that Ucesapp(pn)(X,Y)⊆F(X,Y). Since limn→∞infpn> ,
we have ∞n=(n+)pn <∞, letT ∈Uapp
ces(pn)(X,Y) we get (αn(T))
∞
n=∈ces(pn), and since
ρ((αn(T))∞n=) <∞, let ε∈], ] then there exists a natural number s > such that ρ((αn(T))∞n=s) <
ε
h+δc for somec≥, where δ=max{,
∞
n=s(n+ )
pn}, sinceα
n(T) is de-creasing for eachn∈N, we get
s
n=s+
n+
n
k= αs(T)
pn ≤
s
n=s+
n+
n
k= αn(T)
pn
≤ ∞
n=s
n+
n
k= αk(T)
pn < ε
h+δc, ()
then there existsA∈Fs(X,Y),
rank(A)≤s with
s
n=s+
n+
n
k=
T–A pn
≤
s
n=s+
n+
n
k=
T–A pn
< ε
and
∞
sup
n=s
s
k=
T–A pn
< ε
h+δ, ()
sinceαn(T) =inf{T–A:A∈L(X,Y) and rank(A)≤n}. Then there exists a natural num-ber N > ,AN with rank(AN)≤N andT –AN ≤αN(T). Since αn(T)
n→∞
−→ , then T–AN
N→∞
−→ , so we can take
s
n=
n+
n
k=
T–A pn
< ε
h+δc, ()
since (pn) is an increasing sequence and by using (), (), () and (), we get
d(T,A) =ραn(T–A) ∞
n=
=
s–
n=
n+
n
k=
αk(T–A) pn
+ ∞
n=s
n+
n
k=
αk(T–A) pn ≤ s n= n+
n
k=
T–A pn
+ ∞
n=s
n+
n+s
k=
αk(T–A) pn+s
≤ s n= n+
n
k=
T–A pn
+ ∞
n=s
n+
s–
k=
αk(T–A) + n+
n+s
k=s
αk(T–A) pn ≤ s n= n+
n
k=
T–A pn
+ h–
∞
n=s
n+
s–
k=
αk(T–A) pn
+ ∞
n=s
n+
n+s
k=s
αk(T–A) pn ≤ s n= n+
n
k=
T–A pn
+ h–
∞
n=s
n+
s–
k=
T–A pn
+ ∞
n=s
n+
n
k=
αk+s(T–A) pn ≤ s n= n+
n
k=
T–A pn
+ h–
∞
sup
n=s
s
k=
T–A
pn∞
n=s
n+
pn
+ h– ∞
n=s
n+
n
k= αk(T)
pn
<ε.
Theorem . Let X be a normed space,Y a Banach space and Eρbe a pre modular special
Proof Let (Tm) be a Cauchy sequence inUEappρ (X,Y), then by using Definition . condition
(vii) and sinceUEappρ (X,Y)⊆L(X,Y), we have
ραn(Ti–Tj) ∞
n=
≥ρα(Ti–Tj), , , , . . .
=ρTi–Tj, , , , . . .
≥ζTi–Tjρ(, , , , . . .),
then (Tm) is also Cauchy sequence inL(X,Y). Since the spaceL(X,Y) is a Banach space, then there existsT∈L(X,Y) such thatTm–T
m→∞
−→ and since (αn(Tm))∞n=∈Efor all
m∈N,ρis continuous atθand using Definition .(iii), we have
ραn(T) ∞
n= =ρ
αn(T–Tm+Tm) ∞
n=≤kρ
α[n](Tm–T) ∞
n=+kρ
α[n](Tm) ∞
n=
≤kρTm–T ∞
n=
+kραn(Tm) ∞
n=<ε, for somek≥.
Hence (αn(T))∞n=∈Eas suchT∈U app
Eρ (X,Y).
Corollary . 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 atθ= (, , , . . .)and UappM(X,Y) is complete.
Corollary . Let X be a normed space, Y a Banach space and(pn)be an increasing
sequence of positive real numbers with limn→∞suppn<∞ andlimn→∞infpn> ,then
Ucesapp(pn)(X,Y)is complete.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
NFM gave the idea of the article. AAB carried out the proofs and its application. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Faculty of Science, Ain Shams University, Cairo, Egypt.2Department of Mathematics, Faculty of Science and Arts, King Abdulaziz University (KAU), P.O. Box 80200, Khulais, 21589, Saudi Arabia.
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.
Received: 7 January 2013 Accepted: 5 April 2013 Published: 18 April 2013 References
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