We give sufficient conditions on a special space of sequences defined by Mohamed and Bakery (2013) such that the finite rank operators are dense in the complete space of operators whose approximation numbers belong to this sequence space. Hence, under a few conditions, every compact operator would be approximated by finite rank operators. We apply it on the sequence space defined by Tripathy and Mahanta (2003). Our results match those known for p-absolutely summable sequences of reals.
1. Introduction and Basic Definitions
By ω and L(V,W), we will denote the spaces of all real sequences and all bounded linear operators between two Banach spaces V into W, respectively. In [1], Pietsch, by using the approximation numbers and p-absolutely summable sequences of real numbers, formed the operator ideals. In [2], Mohamed and Bakery have considered the space lM, when M(t)=tp(0<p<∞), which matches especially lp. A subclass U of L={L(V,W)} is an operator ideal if its components verify the following conditions:
The space F(V,W) of all finite rank operators is a subset of U(V,W).
The space U(V,W) is linear.
For two Banach spaces V0 and W0, if T∈L(V0,V), S∈U(V,W), and R∈L(W,W0), then RST∈U(V0,W0). See [3, 4].
An Orlicz function is a function M:[0,∞)→[0,∞) which is convex, positive, nondecreasing, and continuous, where M(0)=0 and limx→∞M(x)=∞. An Orlicz function M is said to satisfy Δ2-condition for all values of x≥0 if there exists a constant k>0, such that M(2x)≤kM(x). Lindenstrauss and Tzafriri [5] used the idea of an Orlicz function to define Orlicz sequence spaces as follows: (1)lM=x∈ω:∃λ>0withρλx=∑k=1∞Mλxk<∞.(lM,x) is a Banach space, where x=inf{λ>0:ρ(x/λ)≤1}. The space lp is an Orlicz sequence space with M(x)=xp for 1≤p<∞.
Remark 1.
For any Orlicz function M, we have M(λx)≤λM(x), for all λ with 0<λ<1.
Let Ps be the class of all subsets of N={0,1,2,…} that do not contain more than s number of elements and let {ϕn} be a nondecreasing sequence of positive reals such that nϕn+1≤(n+1)ϕn, for all n∈N. Tripathy and Mahanta [6] defined and studied the following sequence space: (2)mϕ,M=x=xk∈ω:∃ζ>0withsups≥1,σ∈Ps1ϕs∑k∈σMxkζ<∞,with the norm (3)ρx=infζ>0:sups≥1,σ∈Ps1ϕs∑k∈σMxkζ≤1.
Lemma 2.
(i) lM⊆m(ϕ,M).
(ii) lM=m(ϕ,M) if and only if supsϕs<∞.
As of late, different classes of sequences have been presented using Orlicz functions by Braha [7], Raj and Sharma Sunil [8], Raj et al. [9], and many others ([10–13]).
Definition 3 (see [14, 15]).
A special space of sequences (sss) is a linear space E with the following:
en∈E for all n∈N, where en={0,0,…,1,0,0,…} with 1 appearing at nth place for all n∈N.
E is solid′′.
(x0,x0,x1,x1,…)∈E, if (xn)n∈N∈E.
A premodular (sss) Eρ is a (sss) and there is a function ρ:E→[0,∞[ with the following:
ρ(x)≥0, for each x∈E and ρ(x)=0⇔x=θ, where θ is the zero element of E.
ρ satisfies Δ2-condition.
For each x,y∈E, ρ(x+y)≤k(ρ(x)+ρ(y)) holds for some k≥1.
The space E is ρ-solid; that is, ρ((xn))≤ρ((yn)), whenever |xn|≤|yn|, for all n∈N.
For some numbers k0≥1, the inequality ρ((xn)n∈N)≤ρ((x0,x0,x1,x1,…))≤k0ρ((xn)n∈N) holds.
F¯=Eρ; that is, the set of all finite sequences F is ρ-dense in E.
For each λ>0, there is a constant ξ>0 such that ρ(λ,0,0,0,…)≥ξλρ(1,0,0,0,…).
Condition (ii) says that ρ is continuous at θ. The function ρ defines a metrizable topology in E and the linear space E enriched with this topology is denoted by Eρ.
Definition 4 (see [16]).
Consider the following:(4)UEapp≔UEappV,W,where (5)UEappV,W≔T∈LV,W:αnTn∈N∈E.
Theorem 5 (see [2]).
If E is a (sss), then UEapp is an operator ideal.
We explain some results related to the operator spaces.
2. Main Results
In this part, we give sufficient conditions on E such that the finite rank operators are dense in the complete space of operators UEapp(V,W).
Lemma 6.
If Eρ is a premodular (sss) and (xn)∈Eρ is a decreasing sequence of positive reals, then (6)ρ0,0,0,…,0︷2n,xn,xn+1,xn+2,…≤k0ρ0,0,0,…,0︷n,xn,xn+1,xn+2,….
Proof.
By using Definition 3, conditions (iv) and (v), and since the elements of E are decreasing, we get (7)ρ0,0,0,…,0︷2n,xn,xn+1,xn+2,…≤ρ0,0,0,…,0︷2n,xn,xn,xn+1,xn+1,…≤k0ρ0,0,0,…,0︷n,xn,xn+1,xn+2,….
Theorem 7.
Let Eρ be a premodular (sss); then F(V,W)g¯=UEρapp(V,W), where g(T)=ρ(αn(T)n∈N).
Proof.
To prove that F(V,W)⊆UEapp(V,W), since em∈E for each m∈N, from the linearity of E and T∈F(V,W), then finitely many elements of (αn(T))n∈N are different from zero. Hence, T∈UEapp(V,W). For the other inclusion UEapp(V,W)⊆F(V,W)¯, let T∈UEapp(V,W) and, from the definition of approximation numbers, there is N∈N, N>0, AN with rank(AN)≤N and also (8)T-AN≤2αNT.Since αN(T)→0 as N→∞, then T-AN→0 as N→∞; we have to prove that ρ((αn(T-AN))n∈N)→0 as N→∞, by taking N=8η, where η is a natural number. From Definition 3, condition (iii), we have (9)dT,AN=ραnT-ANn∈N=ρα0T-AN,α1T-AN,…,α8η-1T-AN,0,0,0,…+0,0,0,…,0︷8η,α8ηT-AN,α8η+1T-AN,…,α12η-1T-AN,0,0,0,…+0,0,0,…,0︷12η,α12ηT-AN,α12η+1T-AN,…≤k2ρα0T-AN,α1T-AN,…,α8η-1T-AN,0,0,0,…+ρ0,0,0,…,0︷8η,α8ηT-AN,α8η+1T-AN,…,α12η-1T-AN,0,0,0,…+ρ0,0,0,…,0︷12η,α12ηT-AN,α12η+1T-AN,…=k2I1N+I2η+I3η.Since αn(AN)=0 for n≥N, then (10)αnT-AN≤αn-NT.By using Lemma 6, inequality (10), and Definition 3, condition (v), we get (11)I3η=ρ0,0,0,…,0︷12η,α12ηT-AN,α12η+1T-AN,…≤ρ0,0,0,…,0︷12η,α4ηT,α4η+1T,…≤k0ρ0,0,0,…,0︷6η,α4ηT,α4η+1T,…≤k02ρ0,0,0,…,0︷3η,α4ηT,α4η+1T,…≤k02ρ0,0,0,…,0︷3η,α3ηT,α3η+1T,…=k02ραnTn=3η∞⟶0asη⟶∞.Now, using Lemma 6 and Definition 3, condition (v), we have (12)I2η=ρ0,0,0,…,0︷8η,α8ηT-AN,α8η+1T-AN,…,α12η-1T-AN,0,0,0,…≤k0ρ0,0,0,…,0︷4η,α8ηT-AN,α8η+1T-AN,…,α12η-1T-AN,0,0,0,…≤k0ρα0T-AN,α1T-AN,…,α8η-1T-AN,0,0,0,…=k0I1N.Finally, we have to show that I1(N)→0 as N→∞. Since T∈UEapp(V,W) and ρ is continuous at θ, we have ρ(αk(T))k=n∞→0 as n→∞. Then, for each ε>0, there exists N0(ε) such that for all n≥N0(ε) we have (13)ραkTk=n∞<ε.By taking ε1=ε/3lk for each n≥N0(ε1) and using inequality (13) and Definition 3, conditions (ii) and (iii), then we have (14)I1N=ρα0T-AN,α1T-AN,…,αN-1T-AN,0,0,0,…≤kρα0T-AN,α1T-AN,…,αN0-1T-AN,0,0,0,…+ρ0,0,0,…,0︷N0,αN0T-AN,αN0+1T-AN,…,αN-1T-AN,0,0,0,…≤kρT-AN,T-AN,…,T-AN,0,0,0,…+ρ0,0,0,…,0︷N0,2αNT,2αNT,…,2αNT,0,0,0,…≤kT-ANlρ1,1,1,…,1︷N0,0,0,0,…+2lρ0,0,0,…,0︷N0,αN0T,αN0+1T,…,αNT,0,0,0,…≤kT-ANlk1ε+2lε1,where k1(ε)=ρ(1,1,1,…,1︷N0,0,0,0,…), and since T-AN→0asN→∞, then for each ε>0 there exists N such that T-ANk1(ε)≤ε1, for that we have I1(N)≤k[lε1+2lε1]=3klε1=ε. This completes the proof.
We give here the sufficient conditions on the sequence spaces m(ϕ,M) such that the class of all bounded linear operators between any arbitrary Banach spaces with (αn(T))n∈N in these sequence spaces form an ideal operator; the ideal of the finite rank operators in the class of Banach spaces is dense in Um(ϕ,M)app(V,W).
Theorem 8.
Let M be an Orlicz function satisfying Δ2-condition. Then,
Um(ϕ,M)app is an operator ideal,
F(V,W)¯=Um(ϕ,M)app(V,W).
Proof.
We first prove that the space m(ϕ,M) is a (sss).
(1) Let λ1,λ2∈R and x,y∈m(ϕ,M); then there exist ζ1>0 and ζ2>0 such that (15)sups≥1,σ∈Ps1ϕs∑k∈σMxkζ1<∞,sups≥1,σ∈Ps1ϕs∑k∈σMxkζ2<∞.Let ζ3=max(2|λ1|ζ1,2|λ2|ζ2). Since M is nondecreasing convex function with Δ2-condition, we have (16)∑k∈σMλ1xk+λ2ykζ3≤12∑k∈σMxkζ1+∑k∈σMykζ2.So, we get (17)sups≥1,σ∈Ps1ϕs∑k∈σMλ1xk+λ2ykζ3≤12sups≥1,σ∈Ps1ϕs∑k∈σMxkζ1+sups≥1,σ∈Ps1ϕs∑k∈σMykζ2.Thus, λ1x+λ2y∈m(ϕ,M). Hence, m(ϕ,M) is a linear space over the field of real numbers. Also, since en∈lM and lM⊆m(ϕ,M), we have en∈m(ϕ,M) for all n∈N.
(2) Let x∈ω and y=(yk)k=0∞∈m(ϕ,M) with |xk|≤|yk| for each k∈N; since M is nondecreasing, then we get (18)sups≥1,σ∈Ps1ϕs∑k∈σMxkζ≤sups≥1,σ∈Ps1ϕs∑k∈σMykζ<∞;then x=(xk)k=0∞∈m(ϕ,M).
(3) Let x=(xk)k=0∞∈m(ϕ,M); then we have (19)sups≥1,σ∈Ps1ϕs∑k∈σMxk/2ζ≤2sups≥1,σ∈Ps1ϕs∑k∈σMxkζ<∞;then x=(x[k/2])k=0∞∈m(ϕ,M).
Finally, we have proved that the space m(ϕ,M) with ρ(x) is a premodular (sss).
(i) Clearly, ρ(x)≥0 for all x∈m(ϕ,M) and ρ(x)=0⇔x=θ.
(ii) Let λ∈R and x∈m(ϕ,M); then for λ≠0 we have (20)ρλx=infζ>0:sups≥1,σ∈Ps1ϕs∑k∈σMλxkζ≤1=infλμ>0:sups≥1,σ∈Ps1ϕs∑k∈σMxkμ≤1,where μ=ζ/|λ|. Thus, ρ(λx)=|λ|inf{μ>0:sups≥1,σ∈Ps1/ϕs∑k∈σM(|xk|/μ)≤1}=|λ|ρ(x).
Also, for λ=0, we have ρ(λx)=λρ(x)=0.
(iii) Let x,y∈m(ϕ,M); then there exist ζ1>0 and ζ2>0 such that (21)sups≥1,σ∈Ps1ϕs∑k∈σMxkζ1≤1,sups≥1,σ∈Ps1ϕs∑k∈σMxkζ2≤1.Let ζ=ζ1+ζ2, and since M is nondecreasing and convex, then we have (22)sups≥1,σ∈Ps1ϕs∑k∈σMxk+ykζ≤sups≥1,σ∈Ps1ϕs∑k∈σMxk+ykζ1+ζ2≤sups≥1,σ∈Ps1ϕs∑k∈σζ1ζ1+ζ2Mxkζ1+ζ2ζ1+ζ2Mykζ2≤ζ1ζ1+ζ2sups≥1,σ∈Ps1ϕs∑k∈σMxkζ1+ζ2ζ1+ζ2∑k∈σMykζ2≤1.Since ζ’s are nonnegative, we have (23)ρx+y=infζ>0:sups≥1,σ∈Ps1ϕs∑k∈σMxk+ykζ≤1≤infζ1>0:sups≥1,σ∈Ps1ϕs∑k∈σMxkζ1≤1+infζ2>0:sups≥1,σ∈Ps1ϕs∑k∈σMykζ2≤1=ρx+ρy.
(iv) Let |xk|≤|yk| for each k∈N, and since M is nondecreasing, then we get (24)sups≥1,σ∈Ps1ϕs∑k∈σMxkζ≤sups≥1,σ∈Ps1ϕs∑k∈σMykζ;thus, (25)infζ>0:sups≥1,σ∈Ps1ϕs∑k∈σMxkζ≤infζ>0:sups≥1,σ∈Ps1ϕs∑k∈σMykζ.So, ρ(x)≤ρ(y).
(v) Since (26)sups≥1,σ∈Ps1ϕs∑k∈σMxk/2ζ≤2sups≥1,σ∈Ps1ϕs∑k∈σMxkζ,we have (27)infζ>0:sups≥1,σ∈Ps1ϕs∑k∈σMxk/2ζ≤2infζ>0:sups≥1,σ∈Ps1ϕs∑k∈σMxkζ.So, ρ((xk))≤ρ((x[k/2]))≤2ρ((xk)).
(vi) For each x=(xk)k=0∞∈m(ϕ,M), then (28)ρxkk=0∞=infζ>0:sups≥1,σ∈Ps1ϕs∑k∈σMxk/2ζ<∞;we can find t∈N such that ρ((xk)k=t∞)<∞. This means the set of all finite sequences is ρ-dense in m(ϕ,M).
(vii) For any λ>0, there exists a constant ζ∈]0,1] such that (29)ρλ,0,0,0,…≥ζλρ1,0,0,0,….By using Theorems 5 and 7, the proof follows.
As special cases of the above theorem, we obtain the following corollaries.
Corollary 9.
If supsϕs<∞, one gets that
UlMapp is an operator ideal,
F(V,W)¯=UlMapp(V,W).
Corollary 10.
If supsϕs<∞ and M(t)=tp with 0<p<∞, one gets that
Ulpapp is an operator ideal,
F(V,W)¯=Ulpapp(V,W). See [1].
Theorem 11.
If Eρ is a premodular (sss), then UEρapp(V,W) is complete.
Proof.
Let (Tm) be a Cauchy sequence in UEρapp(V,W); then, by using Definition 3, condition (vii), and since UEρapp(V,W)⊆L(V,W), we get (30)ραnTi-Tjn∈N≥ρα0Ti-Tj,0,0,0,…=ρTi-Tj,0,0,0,…≥ξTi-Tjρ1,0,0,0,…;then (Tm) is also a Cauchy sequence in L(V,W). Since the space L(V,W) is a Banach space, then there exists T∈L(V,W) such that Tm-T→0, as m→∞, and since (αn(Tm))n∈N∈E, for each m∈N, then from Definition 3, conditions (iii) and (v), and since ρ is continuous at θ, we have (31)ραnTn∈N=ραnT-Tm+Tmn∈N≤kραn/2T-Tmn∈N+kραn/2Tmn∈N≤kρTm-Tn∈N+kραnTmn∈N<ε;we get (αn(T))n∈N∈E, and then T∈UEρapp(V,W). This finishes the proof.
By applying Theorem 11 on m(ϕ,M), we can easily conclude the next corollaries.
Corollary 12.
Pick up an Orlicz function M which satisfies Δ2-condition. Then, M is continuous from the right at 0 and Um(ϕ,M)app(V,W) is complete.
Corollary 13.
Pick up an Orlicz function M which satisfies Δ2-condition with supsϕs<∞. Then, M is continuous from the right at 0 and UlMapp(X,Y) is complete.
Corollary 14.
Ulpapp(V,W) is complete if M(t)=tp and p∈(0,∞).
Competing Interests
The author declares that he has no competing interests.
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