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Research article

The new class Lz,p,E of s type operators

  • Received: 09 April 2019 Accepted: 02 June 2019 Published: 05 July 2019
  • MSC : 47B06, 47B37, 47L20

  • The purpose of this study is to introduce the class of s-type Z(u,v;lp(E)) operators, which we denote by Lz,p,E(X,Y), we prove that this class is an operator ideal and quasi-Banach operator ideal by a quasi-norm defined on this class. Then we define classes using other examples of s-number sequences. We conclude by investigating which of these classes are injective, surjective or symmetric.

    Citation: Pınar Zengin Alp, Emrah Evren Kara. The new class Lz,p,E of s type operators[J]. AIMS Mathematics, 2019, 4(3): 779-791. doi: 10.3934/math.2019.3.779

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  • The purpose of this study is to introduce the class of s-type Z(u,v;lp(E)) operators, which we denote by Lz,p,E(X,Y), we prove that this class is an operator ideal and quasi-Banach operator ideal by a quasi-norm defined on this class. Then we define classes using other examples of s-number sequences. We conclude by investigating which of these classes are injective, surjective or symmetric.


    In this study, the set of all natural numbers is represented by N and the set of all nonnegative real numbers is represented by R+.

    If the dimension of the range space of a bounded linear operator is finite, it is called a finite rank operator [1].

    Throughout this study, X and Y denote real or complex Banach spaces. The space of all bounded linear operators from X to Y is denoted by B(X,Y) and the space of all bounded linear operators from an arbitrary Banach space to another arbitrary Banach space is denoted by B.

    The theory of operator ideals is a very important field in functional analysis. The theory of normed operator ideals first appeared in 1950's in [2]. In functional analysis, many operator ideals are constructed via different scalar sequence spaces. An s- number sequence is one of the most important examples of this. The definition of s- numbers goes back to E. Schmidt [3], who used this concept in the theory of non-selfadjoint integral equations. In Banach spaces there are many different possibilities of defining some equivalents of s- numbers, namely Kolmogorov numbers, Gelfand numbers, approximation numbers, and several others. In the following years, Pietsch give the notion of s- number sequence to combine all s- numbers in one definition [4, 5, 6].

    A map

    S:K(sr(K))

    which assigns a non-negative scalar sequence to each operator is called an s-number sequence if for all Banach spaces X,Y,X0 and Y0 the following conditions are satisfied:

    (i) K=s1(K)s2(K)0, for every KB(X,Y),

    (ii) sp+r1(L+K)sp(L)+sr(K) for every L,KB(X,Y) and p,rN,

    (iii) sr(MLK)Msr(L)K for all MB(Y,Y0), LB(X,Y) and  KB(X0,X), where X0,Y0 are arbitrary Banach spaces,

    (iv) If rank(K)r, then sr(K)=0,

    (v) sn1(In)=1, where In is the identity map of n-dimensional Hilbert space ln2 to itself [7].

    sr(K) denotes the rth snumber of the operator K.

    Approximation numbers are frequently used examples of s-number sequence which is defined by Pietsch. ar(K), the r-th approximation number of a bounded linear operator is defined as

    ar(K)=inf{ KA:AB(X,Y)rank(A)<r},

    where KB(X,Y) and rN [4]. Let KB(X,Y) and rN. The other examples of s-number sequences are given in the following, namely Gelfand number (cr(K)), Kolmogorov number (dr(K)), Weyl number (xr(K)), Chang number (yr(K)), Hilbert number (hr(K)), etc. For the definitions of these sequences we refer to [1].

    In the sequel there are some properties of snumber sequences.

    When any isometric embedding JB(Y,Y0) is given and an s-number sequence s=(sr) satisfies sr(K)=sr(JK) for all KB(X,Y) the s-number sequence is called injective [8,p.90].

    Proposition 1. [8,p.90-94] The number sequences (cr(K)) and (xr(K)) are injective.

    When any quotient map SB(X0,X) is given and an s-number sequence s=(sr) satisfies sr(K)=sr(KS) for all KB(X,Y) the s-number sequence is called surjective [8,p.95].

    Proposition 2. [8,p.95] The number sequences (dr(K)) and (yr(K)) are surjective.

    Proposition 3. [8,p.115] Let KB(X,Y). Then the following inequalities hold:

    i) hr(K)xr(K)cr(K)ar(K) and

    ii) hr(K)yr(K)dr(K)ar(K).

    Lemma 1. [5] Let S,KB(X,Y), then |sr(K)sr(S)|KS for r=1,2,.

    Let ω be the space of all real valued sequences. Any vector subspace of ω is called a sequence space.

    In [9] the space Z(u,v;lp) is defined by Malkowsky and Savaş as follows:

    Z(u,v;lp)={xω:n=1|unnk=1vkxk|p<}

    where 1<p< and u=(un) and v=(vn) are positive real numbers.

    The Cesaro sequence space cesp is defined as ([10,11,19])

    cesp={x=(xk)ω:n=1(1nnk=1|xk|)p<},  1<p<.

    If an operator KB(X,Y) satisfies n=1(an(K))p< for 0<p<, K is defined as an lp type operator in [4] by Pietsch. Afterwards ces-p type operators which is a new class obtained via Cesaro sequence space is introduced by Constantin [12]. Later on Tita in [14] proved that the class of lp type operators and ces-p type operators coincide.

    In [15], ς(s)p, the class of stype Z(u,v;lp) operators is given. For more information about sequence spaces and operator ideals we refer to [1,13,16,18,20].

    Let X, the dual of X, be the set of continuous linear functionals on X. The map xy:XY is defined by

    (xy)(x)=x(x)y

    where xX, x X and yY.

    A subcollection  of B is said to be an operator ideal if for each component (X,Y)= B(X,Y) the following conditions hold:

    (i) if x X, yY, then xy(X,Y),

    (ii) if L,K(X,Y), then L+K(X,Y),

    (iii) if L(X,Y), KB(X0,X) and MB(Y,Y0), then MLK(X0,Y0) [6].

    Let be an operator ideal and ρ: R+ be a function on . Then, if the following conditions hold:

    (i) if x X, yY, then ρ(xy)=xy;

    (ii) if C1 such that ρ(L+K)C[ρ(L)+ρ(K)];

    (iii) if L(X,Y),KB(X0,X) and MB(Y,Y0), then ρ(MLK)Mρ(L)K,

    ρ is said to be a quasi-norm on the operator ideal [6].

    For special case C=1, ρ is a norm on the operator ideal .

    If ρ is a quasi-norm on an operator ideal , it is denoted by [,ρ]. Also if every component (X,Y) is complete with respect to the quasi-norm ρ, [,ρ] is called a quasi-Banach operator ideal.

    Let [,ρ] be a quasi-normed operator ideal and JB(Y,Y0) be a isometric embedding. If for every operator KB(X,Y) and JK(X,Y0) we have K(X,Y) and ρ(JK)=ρ(K), [,ρ] is called an injective quasi-normed operator ideal. Furthermore, let [,ρ] be a quasi-normed operator ideal and SB(X0,X) be a quotient map. If for every operator KB(X,Y) and KS(X0,Y) we have K(X,Y) and ρ(KS)=ρ(K), [,ρ] is called an surjective quasi-normed operator ideal [6].

    Let K be the dual of K. An s number sequence is called symmetric (respectively, completely symmetric) if for all KB, sr(K)sr(K) (respectively, sr(K)=sr(K)) [6].

    Lemma 2. [6] The approximation numbers are symmetric, i.e., ar(K)ar(K) for KB.

    Lemma 3. [6] Let KB. Then

    cr(K)=dr(K)   and   cr(K)dr(K).

    In addition, if K is a compact operator then cr(K)=dr(K).

    Lemma 4. [8] Let KB. Then

    xr(K)=yr(K)   and   yr(K)=xr(K)

    .

    The dual of an operator ideal is denoted by and it is defined as [6]

    (X,Y)={KB(X,Y):K(Y,X)}

    .

    An operator ideal is called symmetric if and is called completely symmetric if = [6].

    Let E=(En) be a partition of finite subsets of the positive integers which satisfies

    maxEn<minEn+1

    for nN+. In [21] Foroutannia defined the sequence space lp(E) by

    lp(E)={x=(xn)ω:n=1|jEnxj|p<},        (1p<)

    with the seminorm ||p,E, which defined as:

    |x|p,E=(n=1|jEnxj|p)1p.

    For example, if En={3n2,3n1,3n} for all n, then x=(xn)lp(E) if and only if n=1|x3n2+x3n1+x3n|p<. It is obvious that ||p,E is not a norm, since we have |x|p,E=0 while x=(1,1,0,0,) and En={3n2,3n1,3n} for all n. For the particular case En={n} for nN+ we get lp(E)=lp and |x|p,E=xp.

    For more information about block sequence spaces, we refer the reader to [17,22,23,24,25].

    Let u=(un) and v=(vn) be positive real number sequences. In this section, by replacing lp with lp(E) we get the sequence space Z(u,v;lp(E)) defined as follows:

    Z(u,v;lp(E))={xω:n=1|unnk=1jEkvjxj|p<}.

    An operator KB(X,Y) is in the class of s-type Z(u,v;lp(E)) if

    n=1(unnk=1jEkvjsj(K))p<,  (1<p<).

    The class of all s-type Z(u,v;lp(E)) operators is denoted by Lz,p,E(X,Y).

    In particular case if En={n} for n=1,2,, then the class Lz,p,E(X,Y) reduces to the class ς(s)p.

    Conditions used in Theorem 1 hold throughout the remainder of the paper.

    Theorem 1. Fix 1<p<. If n=1(un)p< and M>0 is such that v2k1+v2kMvk,M>0 for all kN, then Lz,p,E is an operator ideal.

    Proof. Let xX and yY. Since the rank of the operator xy is one, sn(xy)=0 for n2. By using this fact

    n=1(unnk=1jEkvjsj(xy))p=n=1(un)p(v1s1(xy))p=n=1(un)p(v1)pxyp=n=1(un)p(v1)pxpyp<.

    Therefore xyLz,p,E(X,Y).

    Let L,K Lz,p,E(X,Y). Then

    n=1(unnk=1jEkvjsj(L))p<, n=1(unnk=1jEkvjsj(K))p<.

    To show that L+KLz,p,E(X,Y), let us begin with

    nk=1jEkvjsj(L+K)nk=1(jEkv2j1s2j1(L+K)+jEkv2js2j(L+K))nk=1jEk(v2j1+v2j)s2j1(L+K)Mnk=1jEkvj(sj(L)+sj(K))Mnk=1(jEkvjsj(L)+jEkvjsj(K)).

    By using Minkowski inequality we get;

    (n=1(unnk=1(jEkvjsj(L+K)))p)1pM(n=1(unnk=1(jEkvjsj(L)+jEkvjsj(K)))p)1pM[(n=1(unnk=1jEkvjsj(L))p)1p+(n=1(unnk=1jEkvjsj(K))p)1p]<.

    Hence L+KLz,p,E(X,Y).

    Let MB(Y,Y0), LLz,p,E(X,Y) and KB(X0,X). Then,

    n=1(unnk=1jEkvjsj(MLK))pn=1(unnk=1jEkMKvjsj(L))pMpKp(n=1(unnk=1jEkvjsj(L))p)<.

    So MLKLz,p,E(X0,Y0).

    Therefore Lz,p,E(X,Y) is an operator ideal.

    Theorem 2. Kz,p,E=(n=1(unnk=1jEkvjsj(K))p)1p(n=1(un)p)1pv1 is a quasi-norm on the operator ideal Lz,p,E.

    Proof. Let xX and yY. Since the rank of the operator xy is one, sn(xy)=0 for n2. Then

    (n=1(unnk=1jEkvjsj(xy))p)1p(n=1(un)p)1pv1=((n=1(un)p)vp1xyp)1p(n=1(un)p)1pv1=xy=xy .

    Therefore xyz,p,E=xy.

    Let L,KLz,p,E(X,Y). Then

    nk=1jEkvjsj(L+K)nk=1jEkv2j1s2j1(L+K)+jEnv2js2j(L+K)nk=1jEk(v2j1+v2j)s2j1(L+K)Mnk=1jEkvj(sj(L)+sj(K)).

    By using Minkowski inequality we get;

    (n=1(unnk=1jEkvjsj(L+K))p)1p(n=1(Munnk=1jEkvj(sj(L)+sj(K)))p)1pM[(n=1(unnk=1jEkvjsj(L))p)1p+(n=1(unnk=1jEkvjsj(K))p)1p].

    Hence

    L+Kz,p,EM(Sz,p,E+Kz,p,E).

    Let MB(Y,Y0), LLz,p,E(X,Y) and KB(X0,X)

    (n=1(unnk=1jEkvjsj(MLK))p)1p(n=1(unnk=1jEkMKvjsj(L))p)1pMK(n=1(unnk=1jEkvjsj(L))p)1p<
    MLKz,p,EMKLz,p,E.

    Therefore Kz,p,E is a quasi-norm on Lz,p,E.

    Theorem 3. Let 1<p<. [Lz,p,E(X,Y),Kz,p,E] is a quasi-Banach operator ideal.

    Proof. Let X,Y be any two Banach spaces and 1p<. The following inequality holds

    Kz,p,E=(n=1(unnk=1jEkvjsj(K))p)1p(n=1(un)p)1pv1K

    for KLz,p,E(X,Y).

    Let (Km) be Cauchy in Lz,p,E(X,Y). Then for every ε>0 there exists n0N such that

    KmKlz,p,E<ε (2.1)

    for all m,ln0. It follows that

    KmKlKmKlz,p,E<ε.

    Then (Km) is a Cauchy sequence in B(X,Y). B(X,Y) is a Banach space since Y is a Banach space. Therefore KmK0 as m for some KB(X,Y). Now we show that KmKz,p,E0 as m for KLz,p,E(X,Y).

    The operators KlKm, KKm are in the class B(X,Y) for Km,Kl,KB(X,Y).

    |sn(KlKm)sn(KKm)|KlKm(KKm)=KlK.

    Since KlK as l we obtain

    sn(KlKm)sn(KKm) (2.2)

    It follows from (2.1) that the statement

    KmKlz,p,E=(n=1(unnk=1jEkvjsj(KmKl))p)1p(n=1(un)p)1pv1<ε

    holds for all m,ln0. We obtain from (2.2) that

    (n=1(unnk=1jEkvjsj(KmK))p)1p(n=1(un)p)1pv1ε.

    Hence we have

    KmKz,p,E<ε      for all mn0.

    Finally we show that KLz,p,E(X,Y).

    n=1(unnk=1jEkvjsj(K))pn=1(unnk=1jEkv2j1s2j1(K)+unnk=1jEkv2js2j(K))pn=1(unnk=1jEk(v2j1+v2j)s2j1(KKm+Km))pMn=1(unnk=1jEkvj(sj(KKm)+sj(Km)))p

    By using Minkowski inequality; since KmLz,p,E(X,Y) for all m and KmKz,p,E0 as m, we have

    M(n=1(unnk=1jEkvj(sj(KKm)+sj(Km)))p)1pM[(n=1(unnk=1jEkvjsj(KKm))p)1p+(n=1(unnk=1jEkvjsj(Km))p)1p]<

    which means KLz,p,E(X,Y).

    Definition 1. Let μ=(μi(K)) be one of the sequences s=(sn(K)), c=(cn(K)), d=(dn(K)), x=(xn(K)), y=(yn(K)) and h=(hn(K)). Then the space L(μ)z,p,E generated via μ=(μi(K)) is defined as

    L(μ)z,p,E(X,Y)={KB(X,Y):n=1(unnk=1jEkvjμj(K))p<,(1<p<)}.

    And the corresponding norm K(μ)z,p,E for each class is defined as

    K(μ)z,p,E=(n=1(unnk=1jEkvjμj(K))p)1p(n=1(un)p)1pv1.

    Proposition 4. The inclusion L(a)z,p,EL(a)z,q,E holds for 1<pq<.

    Proof. Since lplq for 1<pq< we have L(a)z,p,EL(a)z,q,E.

    Theorem 4. Let 1<p<. The quasi-Banach operator ideal [L(s)z,p,E,K(s)z,p,E] is injective, if the sequence sn(K) is injective.

    Proof. Let 1<p< and KB(X,Y) and JB(Y,Y0) be any isometric embedding. Suppose that JKL(s)z,p,E(X,Y0). Then

    n=1(unnk=1jEkvjsj(JK))p<

    Since s=(sn) is injective, we have

    sn(K)=sn(JK) for all KB(X,Y),n=1,2,. (2.3)

    Hence we get

    n=1(unnk=1jEkvjsj(K))p=n=1(unnk=1jEkvjsj(JK))p<

    Thus KL(s)z,p,E(X,Y) and we have from (2.3)

    JK(s)z,p,E=(n=1(unnk=1jEkvjsj(JK))p)1p(n=1(un)p)1pv1=(n=1(unnk=1jEkvjsj(K))p)1p(n=1(un)p)1pv1=K(s)z,p,E

    Hence the operator ideal [L(s)z,p,E,K(s)z,p,E] is injective.

    Conclusion 1. [8,p.90-94] Since the number sequences (cn(K)) and (xn(K)) are injective, the quasi-Banach operator ideals [L(c)z,p,E,K(c)z,p,E] and [L(x)z,p,E,K(x)z,p,E] are injective.

    Theorem 5. Let 1<p<. The quasi-Banach operator ideal [L(s)z,p,E,K(s)z,p,E] is surjective, if the sequence (sn(K)) is surjective.

    Proof. Let 1<p< and KB(X,Y) and SB(X0,X) be any quotient map. Suppose that KSL(s)z,p,E(X0,Y). Then

    n=1(unnk=1jEkvjsj(KS))p<.

    Since s=(sn) is surjective, we have

    sn(K)=sn(KS) for all KB(X,Y),n=1,2,. (2.4)

    Hence we get

    n=1(unnk=1jEkvjsj(K))p=n=1(unnk=1jEkvjsj(KS))p<.

    Thus KL(s)z,p,E(X,Y) and we have from (2.4)

    KS(s)z,p,E=(n=1(unnk=1jEkvjsj(KS))p)1p(n=1(un)p)1pv1=(n=1(unnk=1jEkvjsj(K))p)1p(n=1(un)p)1pv1=K(s)z,p,E.

    Hence the operator ideal [L(s)z,p,E,K(s)z,p,E]is surjective.

    Conclusion 2. [8,p.95] Since the number sequences (dn(K)) and (yn(K)) are surjective, the quasi-Banach operator ideals [L(d)z,p,E,K(d)z,p,E] and [L(y)z,p,E,K(y)z,p,E] are surjective.

    Theorem 6. Let 1<p<. Then the following inclusion relations holds:

    i L(a)z,p,EL(c)z,p,EL(x)z,p,EL(h)z,p,E

    ii L(a)z,p,EL(d)z,p,EL(y)z,p,EL(h)z,p,E.

    Proof. Let KL(a)z,p,E. Then

    n=1(unnk=1jEkvjsj(K))p<

    where 1<p<. And from Proposition 3, we have;

    n=1(unnk=1jEkvjhj(K))pn=1(unnk=1jEkvjxj(K))pn=1(unnk=1jEkvjcj(K))pn=1(unnk=1jEkvjaj(K))p<

    and

    n=1(unnk=1jEkvjhj(K))pn=1(unnk=1jEkvjyj(K))pn=1(unnk=1jEkvjdj(K))pn=1(unnk=1jEkvjaj(K))p<.

    So it is shown that the inclusion relations are satisfied.

    Theorem 7. For 1<p<, L(a)z,p,E is a symmetric operator ideal and L(h)z,p,E is a completely symmetric operator ideal.

    Proof. Let 1<p<.

    Firstly, we show that L(a)z,p,E is symmetric in other words L(a)z,p,E(L(a)z,p,E) holds. Let KL(a)z,p,E. Then

    n=1(unnk=1jEkvjaj(K))p<.

    It follows from [6, p.152] an(K)an(K) for KB. Hence we get

    n=1(unnk=1jEkvjaj(T))pn=1(unnk=1jEkvjaj(K))p<.

    Therefore K(L(a)z,p,E). Thus L(a)z,p,E is symmetric.

    Now we prove that the equation L(h)z,p,E=(L(h)z,p,E) holds. It follows from [8,p.97] that hn(K)=hn(K) for KB. Then we can write

    n=1(unnk=1jEkvjhj(K))p=n=1(unnk=1jEkvjhj(K))p.
    n=1(unnk=1jEkvjhj(K))p=n=1(unnk=1jEkvjhj(K))p.

    Hence L(h)z,p,E is completely symmetric.

    Theorem 8. Let 1<p<. The equation L(c)z,p,E=(L(d)z,p,E) and the inclusion relation L(d)z,p,E(L(c)z,p,E) holds. Also, for a compact operator K, KL(d)z,p,E if and only if K(L(c)z,p,E).

    Proof. Let 1<p<. For KB it is known from [8] that cn(K)=dn(K) and cn(K)dn(K). Also, when K is a compact operator, the equality cn(K)=dn(K) holds. Thus the proof is clear.

    Theorem 9. L(x)z,p,E=(L(y)z,p,E) and L(y)z,p,E=(L(x)z,p,E) hold for 1<p<.

    Proof. Let 1<p<. For KB we have from [8] that xn(K)=yn(K) and yn(K)=xn(K). Thus the proof is clear.

    The authors would like to thank anonymous referees for their careful corrections and valuable comments on the original version of this paper.

    The authors declare no conflict of interest.



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