Loading [MathJax]/jax/output/SVG/jax.js
Research article

Compact operators on the new Motzkin sequence spaces

  • Received: 12 June 2024 Revised: 21 July 2024 Accepted: 26 July 2024 Published: 15 August 2024
  • MSC : 11B83, 46A45, 46B45, 47B07, 47B37

  • This study aims to construct the BK-spaces p(M) and (M) as the domains of the conservative Motzkin matrix M obtained by using Motzkin numbers. It investigates topological properties, obtains Schauder basis, and then gives inclusion relations. Additionally, it expresses α-, β-, and γ-duals of these spaces and submits the necessary and sufficient conditions of the matrix classes between the described spaces and the classical spaces. In the last part, the characterization of certain compact operators is given with the aid of the Hausdorff measure of non-compactness.

    Citation: Sezer Erdem. Compact operators on the new Motzkin sequence spaces[J]. AIMS Mathematics, 2024, 9(9): 24193-24212. doi: 10.3934/math.20241177

    Related Papers:

    [1] Keun Young Lee, Gwanghyun Jo . The dual of a space of compact operators. AIMS Mathematics, 2024, 9(4): 9682-9691. doi: 10.3934/math.2024473
    [2] Carlos F. Álvarez, Javier Henríquez-Amador, John Millán G., Eiver Rodríguez . On the composition operator with variable integrability. AIMS Mathematics, 2025, 10(2): 2021-2041. doi: 10.3934/math.2025095
    [3] Zhiyu Lin, Xiangxing Tao, Taotao Zheng . Compactness for iterated commutators of general bilinear fractional integral operators on Morrey spaces with non-doubling measures. AIMS Mathematics, 2022, 7(12): 20645-20659. doi: 10.3934/math.20221132
    [4] Lian Hu, Rong Yang, Songxiao Li . Embedding and Volterra integral operators on a class of Dirichlet-Morrey spaces. AIMS Mathematics, 2021, 6(7): 7782-7797. doi: 10.3934/math.2021453
    [5] Ruishen Qian, Xiangling Zhu . Embedding of Qp spaces into tent spaces and Volterra integral operator. AIMS Mathematics, 2021, 6(1): 698-711. doi: 10.3934/math.2021042
    [6] Heng Yang, Jiang Zhou . Compactness of commutators of fractional integral operators on ball Banach function spaces. AIMS Mathematics, 2024, 9(2): 3126-3149. doi: 10.3934/math.2024152
    [7] Taja Yaying, S. A. Mohiuddine, Jabr Aljedani . Exploring the $ q $-analogue of Fibonacci sequence spaces associated with $ c $ and $ c_0 $. AIMS Mathematics, 2025, 10(1): 634-653. doi: 10.3934/math.2025028
    [8] Aydah Mohammed Ayed Al-Ahmadi . Differences weighted composition operators in several variables between some spaces of analytic functions. AIMS Mathematics, 2023, 8(11): 27363-27375. doi: 10.3934/math.20231400
    [9] Feng Qi, Bai-Ni Guo . Several explicit and recursive formulas for generalized Motzkin numbers. AIMS Mathematics, 2020, 5(2): 1333-1345. doi: 10.3934/math.2020091
    [10] Stevo Stević . Note on a new class of operators between some spaces of holomorphic functions. AIMS Mathematics, 2023, 8(2): 4153-4167. doi: 10.3934/math.2023207
  • This study aims to construct the BK-spaces p(M) and (M) as the domains of the conservative Motzkin matrix M obtained by using Motzkin numbers. It investigates topological properties, obtains Schauder basis, and then gives inclusion relations. Additionally, it expresses α-, β-, and γ-duals of these spaces and submits the necessary and sufficient conditions of the matrix classes between the described spaces and the classical spaces. In the last part, the characterization of certain compact operators is given with the aid of the Hausdorff measure of non-compactness.



    All real sequence's space is expressed by ω. Any Λω is called a sequence space. Frequently encountered sequence spaces can be denoted as (space of bounded sequences), c (space of convergent sequences), c0 (space of null sequences), and p (1p<) (space of absolutely p-summable sequences). These are Banach spaces with u=uc=uc0=suprN|ur| and up=(r|ur|p)1p for u=(ur)ω, r|ur|=+r=0|ur| and N={0,1,2,3,...}. Furthermore, bs and cs denote the bounded and convergent series' spaces, respectively.

    A Banach space on which all coordinate functionals κr defined as κr(u)=ur are continuous is named a BK-space. Consider the sequence e(r) whose rth term is 1 and other terms are zero. If each u=(ur)Λω can be expressed uniquely as u=rure(r), then the space Λ satisfies the AK-property.

    Consider the multiplier set (ΛΥ) defined by

    (ΛΥ)={τ=(τr)ω:τu=(τrur)Υ for alluΛ},

    for Λ,Υω. Thus, the α-, β- and γ-duals of Λ are expressed by

    Λα=(Λ1),Λβ=(Λcs) andΛγ=(Λbs).

    Br refers to the rth row of an infinite matrix B=(brs) with real terms. Additionally, if the series is convergent for all rN, (Bu)r=sbrsus is a B-transform of uω. The infinite matrix B is a matrix mapping from Λ to Υ, if BuΥ for all uΛ. The class of all matrix mappings described from Λ to Υ is given by (Λ:Υ). Additionally, B(Λ:Υ) iff BrΛβ and BuΥ for all uΛ. The set

    ΥB={uω:BuΥ}, (1.1)

    is a matrix domain of B in Υ.

    Spaces Λ and Υ, between which a norm-preserving bijection can be defined, are linearly norm isomorphic spaces, and this situation is denoted by ΛΥ.

    One of the impressive number sequences is the integer sequences consisting of Motzkin numbers which are named after Theodore Motzkin [21]. In mathematics, the rth Motzkin number refers to the number of distinct chords that can be drawn between r points on a circle without intersecting. It should be noted that it is not necessary for the chord to touch all points on the circle. The Motzkin numbers Mr (rN) have various applications in geometry, combinatorics, and number theory, and they are represented by the following sequence:

    1,1,2,4,9,21,51,127,323,835,2188,5798,15511,41835,.

    The Motzkin numbers satisfy the recurrence relations

    Mr=Mr1+r2s=0MsMrs2=2r+1r+1Mr1+3r3r+2Mr2.

    Another relation provided by the Motzkin numbers is given below:

    Mr+2Mr+1=rs=0MsMrs, for r0. (1.2)

    Furthermore, there are two other relations between Motzkin and Catalan numbers Cr, presented by

    Mr=r2s=0(r2s)Cs andCr+1=rs=0(rs)Ms,

    where . is the floor function.

    The generating function m(u)=+r=0Mrur of the Motzkin numbers holds

    u2+[m(u)]2+(u1)m(u)+1=0

    and is described by

    m(u)=1u12u3u22u2.

    The expression on Motzkin numbers with the help of integral function is as follows:

    Mr=2ππ0sin2u(2cosu+1)rdu.

    They have the asymptotic behavior

    Mr12π(3r)323r,r.

    Moreover, it is satisfied from Aigner [1] that

    limr+Mr+1Mr=3.

    The idea of obtaining sequence spaces with infinity matrices created with the help of integer sequences, such as Schröder [6,7], Mersenne [8], Catalan [12,13,16], Fibonacci [14,15], Lucas [17,18], Bell [19], Padovan [28], and Leonardo [29], has been used by various authors for this purpose. In this context, sources such as summability theory, sequence spaces, matrix domain studies and the necessary basic concepts can be specified [2,4,5,9,11,24].

    The Motzkin matrix M=(mrs)r,sN constructed with the help of Motzkin numbers and relation (1.2) by Erdem et al. [10] can be written follows:

    mrs:={MsMrsMr+2Mr+1, if 0sr,0, if s>r. (1.3)

    It is possible to state the Motzkin matrix more clearly as follows:

    M:=[M0M0M2M100000M0M1M3M2M1M0M3M20000M0M2M4M3M1M1M4M3M2M0M4M3000M0M3M5M4M1M2M5M4M2M1M5M4M3M0M5M400M0M4M6M5M1M3M6M5M2M2M6M5M3M1M6M5M4M0M6M50M0M5M7M6M1M4M7M6M2M3M7M6M3M2M7M6M4M1M7M6M5M0M7M6].

    Furthermore, it is also known from [10] that the Motzkin matrix M is conservative, that is, M(c:c) and the inverse M1=(m1rs) of the Motzkin matrix M is

    m1rs:={(1)rsMs+2Ms+1Mrπrs, if 0sr,0, if s>r, (1.4)

    where π0=0 and

    πr=|M1M0000M2M1M000M3M2M1M00MrMr1Mr2Mr3M1|,

    for all rN{0}. From its definition, it is clear that M is a triangle. Furthermore, M-transform of a sequence u=(us) is stated as

    νr:=(Mu)r=1Mr+2Mr+1rs=0MsMrsus,(rN). (1.5)

    The idea of constructing new normed sequence spaces as domains of special infinite matrices, as an application of summability theory to sequence spaces, has appeared as a favorite research area in recent years. Creating these infinite matrices with the help of special number sequences and thus obtaining new normed sequence spaces and also examining some of properties (e.g., completeness, inclusion relations, Schauder basis, duals, matrix transformations, compact operators, and core theorems) is newer topic of study since Kara and Başarır [14].

    The primary research question of this study is whether it is possible to obtain new normed sequence spaces as the domain of the conservative Motzkin matrix obtained with the help of Motzkin numbers and to examine some of the properties just mentioned above on these spaces.

    In this study, two new sequence spaces are obtained as domains of the Motzkin matrix on p (1p<+) and . Subsequently, some algebraic and topological properties, inclusion relations, basis, duals, and matrix transformations of these spaces are presented. In the last section, compactness criteria of some matrix operators defined on these spaces are investigated.

    In this part, we introduce the BK-sequence spaces p(M) and (M) with the help of the Motzkin matrix which are linearly isomorphic to p and , respectively. Finally, the Schauder basis of p(M) and the inclusion relations of the spaces are presented.

    Now, we may present the Motzkin sequence spaces p(M) and (M) as follows:

    p(M)={u=(us)ω:r|1Mr+2Mr+1rs=0MsMrsus|p<},

    and

    (M)={u=(us)ω:suprN|1Mr+2Mr+1rs=0MsMrsus|<},

    for 1p<+. Then, p(M) and (M) can be rewritten as p(M)=(p)M and (M)=()M with the notation (1.1). The matrix domain ΥM is a Motzkin sequence space for each normed sequence space Υ.

    It should be noted that BK-spaces have a significant role in summability theory. For instance, the matrix operators between BK-spaces are continuous. Additionally, there is a useful method for the characterizations of compact linear operators between the spaces as an application of the Hausdorf measure of non-compactness.

    Now we can show that the newly defined spaces are BK-spaces:

    Theorem 2.1. p(M) and (M) are BK-spaces with

    up(M)=(r|1Mr+2Mr+1rs=0MsMrsus|p)1p,

    and

    u(M)=suprN|1Mr+2Mr+1rs=0MsMrsus|,

    respectively.

    Proof. In the proof of this theorem, it will be used that p and are BK-spaces. It is known from Wilansky [27] that ΥB is BK-space with uΥB=BuΥ if B is triangle and Υ is a BK-space. Consequently, p(M) and (M) are BK-spaces with the norms .p(M) and .(M), respectively.

    Theorem 2.2. p(M)p and (M).

    Proof. In order to show that two spaces are linearly norm isomorphic, there must be a linear norm preserving bijection between them. The mapping K:p(M)p, K(u)=Mu is linear and since K(u)=0u=0, K is injective. Consider the sequences ν=(νs)p and u=(us)ω with

    us=si=0(1)siMi+2Mi+1Msπsiνi.(sN). (2.1)

    For the equation

    (Mu)r=1Mr+2Mr+1rs=0MsMrsus=1Mr+2Mr+1rs=0MsMrssi=0(1)siMi+2Mi+1Msπsiνi=1Mr+2Mr+1ri=0(ris=0(1)sMrsiπs)(Mi+2Mi+1)νi=1Mr+2Mr+1((1)0M0π0)(Mr+2Mr+1)νr=νr,

    for ris=0(1)sMrsiπs=0, if ri, then K is surjective. Additionally, since the relation up(M)=Mup holds, then K keeps the norm.

    The proof can be done similarly for the other spaces.

    As a result of the isomorphism between the mentioned spaces, connections can be established between some properties. However, since the new concepts presented here may enable different perspectives and open a new door to generalizations, it is useful to express some basic theorems for newly defined spaces.

    For a normed sequence space (Λ,.) and (ηr)Λ, (ηr) is a Schauder basis for Λ if for any uΛ, there is a unique sequence (σr) of scalars such that

    urs=0σsηs0,

    as r+. This can be stated as u=sσsηs.

    The inverse image of the basis (e(s))sN of p becomes the basis of p(M) (1p<+) since K:p(M)p is an isomorphism in Theorem 2.2. Therefore, it will be given the following result without proof.

    Theorem 2.3. The set η(s)=(η(s)r)p(M) expressed by

    η(s)r:={(1)rsMs+2Ms+1Mrπrs,if0sr,0,ifs>r,

    is a Schauder basis for p(M) and the unique representation of any up(M) is stated as u=sσsη(s) for 1p<+ and σs=(Mu)s.

    It is worth noting that since every normed linear space with a Schauder basis is separable, the sequence space p(M) is seperable.

    Theorem 2.4. The inclusion p(M)˜p(M) strictly holds for 1p<˜p<+.

    Proof. For the first part, it is sufficient to show that every element taken from p(M) is in ˜p(M). Let us take u=(us)p(M) such that Mup. Furthermore, it is known that p˜p for 1p<˜p<+. Then, Mu˜p and u=(us)˜p(M).

    If ˜ν=M˜u˜pp, then the rest of the proof is completed.

    Theorem 2.5. (M).

    Proof. For the proof, it is necessary to show that every element taken from is in (M). For u=(us), it is clear that

    u(M)=suprN|1Mr+2Mr+1rs=0MsMrsus|usuprN|1Mr+2Mr+1rs=0MsMrs|=u<+.

    Thus, u(M).

    In third part of the article, duals of new spaces will be found.

    The idea of dual space plays an important role in the representation of linear functionals and the characterization of matrix transformations between sequence spaces. Consider that ξ{α,β,γ}. In that case,

    ξ1=,ξ=cξ0=cξ=1 and ξp=q, where 1<p,q<+ with q=pp1.

    Additionally, for any Λω, ΛαΛβΛγ. If (brs)sNΛβ for all rN, then, the B-transform (Bu)r=sbrsus of any sequence u=(us)Λ is convergent for an infinite matrix B.

    Now, consider the conditions (3.1)–(3.12) as

    supsNr|brs|<, (3.1)
    supsNr|brs|p<, (3.2)
    supr,sN|brs|<, (3.3)
    limr+brs exists for all sN, (3.4)
    limr+brs=0, (3.5)
    supEFs|rEbrs|q<, (3.6)
    suprNs|brs|q<, (3.7)
    supEFr|sEbrs|<, (3.8)
    supEFr|sEbrs|p<, (3.9)
    suprNs|brs|<, (3.10)
    limr+s|brs|=s|limr+brs|, (3.11)
    limr+s|brs|=0, (3.12)

    where, the FN is finite and 1<p<+. It can be now presented the table created using [26] and characterizing some matrix classes:

    Let us consider the sets ϖ1ϖ7 which will be used to compute the duals:

    ϖ1={τ=(τs)ω:supEFs|rE(1)rsMs+2Ms+1Mrπrsτr|q<},ϖ2={τ=(τs)ω:supsNr|(1)rsMs+2Ms+1Mrπrsτr|<},ϖ3={τ=(τs)ω:supEFr|sE(1)rsMs+2Ms+1Mrπrsτr|<},ϖ4={τ=(τs)ω:limr+ri=s(1)isMs+2Ms+1Miπisτi exists for each sN},ϖ5={τ=(τs)ω:suprNs|ri=s(1)isMs+2Ms+1Miπisτi|q<},ϖ6={τ=(τs)ω:supr,sN|ri=s(1)isMs+2Ms+1Miπisτi|<},ϖ7={τ=(τs)ω:limr+s|ri=s(1)isMs+2Ms+1Miπisτi|=s|i=s(1)isMs+2Ms+1Miπisτi|}.

    Theorem 3.1. The following statements hold:

    (ⅰ) (p(M))α=ϖ1, (1<p<+),

    (ⅱ) (1(M))α=ϖ2,

    (ⅲ) ((M)α=ϖ3.

    Proof. (ⅰ) From (1.5),

    τrur=τr(rs=0(1)rsMs+2Ms+1Mrπrsνs)=(rs=0(1)rsMs+2Ms+1Mrπrsτr)νs=(Gν)r, (3.13)

    for all rN and up(M), where the infinite matrix G=(grs) can described as

    grs:={(1)rsMs+2Ms+1Mrπrsτs, if 0sr,0, if s>r.

    Then by (3.13), τu=(τrur)1 while up(M) iff Gν1 while νp. Thus, τ(p(M))α iff G(p:1). From Table 1, (p(M))α=ϖ1 for 1<p<+.

    Table 1.  Characterizations of (Λ:Υ), where Λ,Υ{1,p,,c,c0} and 1<p<+.
    (Λ↓:Υ) 1 p c c0
    1 (3.1) (3.2) (3.3) (3.3), (3.4) (3.3), (3.5)
    p (3.6) (3.7) (3.4), (3.7) (3.5), (3.7)
    (3.8) (3.9) (3.10) (3.4), (3.11) (3.12)
    c (3.8) (3.9) (3.10)
    c0 (3.8) (3.9) (3.10)
    Note: The symbol "" represents the conditions of the classes that are unknown or not interest to this study.

     | Show Table
    DownLoad: CSV

    The proofs of (ⅱ) and (ⅲ) can be seen similarly to the first part through the aid of the conditions of the classes (1:1) and (:1), respectively, from Table 1. Therefore, they are ommitted.

    Theorem 3.2. The following statements hold:

    (ⅰ) (p(M))β=ϖ4ϖ5, (1<p<+),

    (ⅱ) (1(M))β=ϖ4ϖ6,

    (ⅲ) ((M)β=ϖ4ϖ7.

    Proof. (ⅰ) Consider that τ=(τs)ω and up(M) with νp as (1.5). By considering the Eq (2.1), we can see that

    ψr=rs=0τsus=rs=0τs(si=0(1)siMi+2Mi+1Msπsi)νi=rs=0(ri=s(1)isMs+2Ms+1Miπisτi)νs=(Oν)r, (3.14)

    for O=(ors) is expressed by

    ors:={ri=s(1)isMs+2Ms+1Miπisτi,0sr,0,s>r, (3.15)

    for every r,sN. Then, from (3.14), τucs while u=(us)p(M) iff ψ=(ψr)c while νp. In this case, τ(p(M))β iff O(p:c). Consequently, from the conditions of (p:c) in Table 1, the proof is complete.

    The proofs of the (ⅱ) and (ⅲ) can easily be seen similarly to the first part through the aid of the conditions of the classes (1:c) and (:c), respectively, from the Table 1. Therefore, they are ommitted too.

    Theorem 3.3. The following statements hold:

    (ⅰ) (p(M))γ=ϖ5,(1<p<+),

    (ⅱ) (1(M))γ=ϖ6,

    (ⅲ) ((M)γ=ϖ5 with q=1.

    Proof. This can be done similarly with Theorem 3.2 by considering with together the classes (p:), (1:), and (:) from Table 1 with O=(ors) expressed by (3.15).

    This section offers to submit the matrix classes related Motzkin sequence spaces described in this study. The theorem it will be written now forms the fundamental of this section.

    Theorem 4.1. Let us consider the Λ,Υω, infinite matrices H(r)=(h(r)is) and H=(hrs) described as

    h(r)is:={ij=s(1)jsMs+2Ms+1Mjπjsbrj,0si,0,s>i, (4.1)

    and

    hrs=j=s(1)jsMs+2Ms+1Mjπjsbrj, (4.2)

    for all r,sN. In that case, B=(brs)(Λ(M):Υ) if and only if H(r)(Λ:c) and H(Λ:Υ).

    Proof. Let us consider B=(brs)(Λ(M):Υ) and uΛ(M). In that case,

    is=0brsus=is=0brs(sj=0(1)sjMj+2Mj+1Msπsjνj)=is=0(ij=s(1)jsMs+2Ms+1Mjπjsbrj)νs=is=0h(r)isνs, (4.3)

    for all i,rN. Since Bu exists, then H(r)(Λ:c). By passing limit for i+ in the relation (4.3), Bu=Hν. Since BuΥ, HνΥ and so H(Λ:Ψ).

    Conversely, let us suppose that H(r)(Λ:c) and H(Λ:Υ). Then, hrsΛβ which gives us (brs)sN(Λ(M))β for all rN. Hence, Bu exists for all uΛ(M). Therefore, from relation (4.3) for i+, Bu=Hν. Thus B(Λ(M):Υ), which is desired result.

    Corollary 4.2. Consider the infinite matrices H(r)=(h(r)is) and H=(hrs) described with the relations (4.1) and (4.2), respectively. Then, the conditions of the classes (Γ(M):Ψ) can be deduced from Table 2, where Λ{1,p,}, Υ{1,p,,c,c0} and 1<p<+.

    Table 2.  Characterizations of the classes (Λ(M):Υ), where Λ{1,p,}, Υ{1,p,,c,c0} and 1<p<+.
    (Λ(M)↓:Υ) 1 p c c0
    1(M) (3.3)r,(3.4)r(3.1) (3.3)r,(3.4)r(3.2) (3.3)r,(3.4)r(3.3) (3.3)r,(3.4)r(3.3),(3.4) (3.3)r,(3.4)r(3.3),(3.5)
    p(M) (3.4)r,(3.7)r(3.6) (3.4)r,(3.7)r(3.7) (3.4)r,(3.7)r(3.4),(3.7) (3.4)r,(3.7)r(3.5),(3.7)
    (M) (3.4)r,(3.11)r(3.8) (3.4)r,(3.11)r(3.9) (3.4)r,(3.11)r(3.10) (3.4)r,(3.11)r(3.4),(3.11) (3.4)r,(3.11)r(3.12)
    Note: Conditions (λ)r and (λ) represent the condition (λ) hold with the matrices H(r) and H, respectively, for 3.1λ3.12.

     | Show Table
    DownLoad: CSV

    Theorem 4.3. Consider the infinite matrices ˜B=(˜brs) and B=(brs) described with the relation

    ˜brs=rj=0MjMrjMr+2Mr+1bjs. (4.4)

    In that case, B(Λ:Υ(M)) iff ˜B(Λ:Υ) for Λ{1,p,,c,c0} and Υ{1,p,}.

    Proof. Consider that the infinite matrices ˜B and B described with the relation (4.4), Λ{1,p,,c,c0}, and Υ{1,p,}. For any u=(us)Λ,

    s=0˜brsus=rj=0MjMrjMr+2Mr+1s=0bjsus.

    This means that ˜Br(u)=Mr(Bu) for all rN, which implies that BuΥ(M) iff ˜BuΥ for every uΛ. Thus, B(Λ:Υ(M)) if and only if ˜B(Λ:Υ).

    Corollary 4.4. Consider the infinite matrices ˜B=(˜brs) and B=(brs) described with the relation (4.4). In that case, the necessary and sufficient conditions for the classes (Λ:Υ(M)) can be found in Table 3, where Λ{1,p,,c,c0} and Υ{1,p,}.

    Table 3.  Characterizations of (Λ:Υ(M)), where Λ{1,p,,c,c0} and Υ{1,p,}.
    (Λ↓:Υ(M))) 1(M) p(M) (M)
    1 (3.1) (3.2) (3.3)
    p (3.6) (3.7)
    (3.8) (3.9) (3.10)
    c (3.8) (3.9) (3.10)
    c0 (3.8) (3.9) (3.10)
    Note: Conditions hold with the matrix ˜B=(˜brs).

     | Show Table
    DownLoad: CSV

    Measures of non-compactness play an important role in functional analysis. They are important tools in metric fixed point theory, the theory of operator equations in Banach spaces, and the characterizations of classes of compact operators. They are also applied in the studies of various kinds of differential and integral equations. For instance, the characterization of compact operators between BK-spaces benefits from Hausdorff measure of non-compactness.

    Consider the normed space Λ and the unit sphere DΛ in Λ. The notation uΛ is expressed by

    uΛ=supxDΛ|susxs|,

    for a BK-space ΛΩ and u=(us)ω for all finite sequences' space, Ω, provided that the series is finite and uΛβ.

    Lemma 5.1. [20] The following are satisfied:

    (ⅰ) β=1 and u=u1, u1.

    (ⅱ) β1= and u1=u, u.

    (ⅲ) βp=q and up=uq, uq.

    The set of all bounded linear mappings from Λ to Υ is denoted by C(Λ:Υ).

    Lemma 5.2. [20] There exists KBC(Λ:Υ) as KB(u)=Bu for BK-spaces Λ and Υ, as well as for all uΛ and B(Λ:Υ).

    Lemma 5.3. [20] If B(Λ:Υ), then KB=B(Λ:Υ)=suprNBrΛ< for Υ{c0,c,} and the BK-space ΛΩ.

    The Hausdorff measure of non-compactness of a bounded set P in the metric space Λ is stated with

    χ(P)=inf{ϵ>0:Prj=1Q(uj,nj),ujΛ,nj<ϵ,rN0},

    where Q(uj,nj) is the open ball centred at uj and radius nj for each j=1,2,...,r. In-depth information on the subject can be obtained from [20] and its references.

    Theorem 5.4. [25] Consider that Pp is bounded and mapping Ψn:pp stated by Ψn(u)=(u0,u1,u2,...,un,0,0,...) for all u=(us)p, 1p<+ and nN. Then,

    χ(P)=limn(supuP(IΨn)(u)p),

    for the identity operator I on p.

    A linear mapping K is compact if (K(u)) has a convergent subsequence in Υ for all u=(us)Λ for the Banach spaces Λ and Υ.

    The Hausdorff measure of non-compactness Kχ of K is expressed with Kχ=χ(K(DΛ)). In that case, K is compact iff Kχ=0.

    The studies [3,22,23] can be given as examples of studies on sequence spaces considered in terms of compactness and Hausdorff measure of non-compactness relationship.

    The following results will given for the sequences x=(xs) and y=(ys) which are elements of ω and are attached to each other by the relation

    ys=j=s(1)jsMs+2Ms+1Mjπjsxj, (5.1)

    for all sN.

    Lemma 5.5. Let us consider the sequence x=(xs)(p(M))β for 1p. In that case, y=(ys)q and

    sxsus=sysνs, (5.2)

    for all u=(us)p(M).

    Lemma 5.6. Let us consider the sequence y=(ys) described with relation (5.1). In that case, the following statements hold:

    (ⅰ) x(M)=s|ys|< for all x=(xs)((M))β.

    (ⅱ) x1(M)=sups|ys|< for all x=(xs)(1(M))β.

    (ⅲ) xp(M)=(s|ys|q)1q< for all x=(xs)(p(M))β and 1<p<+.

    Proof. Only a proof of the first part will be given because the other parts are similar.

    (ⅰ) From Lemma 5.5, y=(ys)1 and (5.2) holds for x=(xs)((M))β and for all u=(us)(M). Since u(M)=ν with (1.5), then uD(M) if and only if νD. Thus, we can write the equality x(M)=supuD(M)|sxsus|=supνD|sysνs|=y. By the aid of the Lemma 5.1, it follows that x(M)=y=y1=s|ys|<.

    Lemma 5.7. [22] Considering the BK-space ΛΩ;

    (ⅰ) If B(Λ:), then 0KBχlim suprBrΛ and KB is compact if limrBrΛ=0.

    (ⅱ) If B(Λ:c0), then KBχ=lim suprBrΛ and KB is compact if and only if limrBrΛ=0.

    (ⅲ) If B(Λ:1), then

    limj(supEFjrEBrΛ)KBχ4.limj(supEFjrEBrΛ),

    and KB is compact iff limj(supEFjrEBrΛ)=0, where F represents the family of all finite subsets of N and Fj is the subcollection of F consisting of subsets of N with elements that are greater than j.

    The matrices H=(hrs) and B=(brs) connected with (4.2) will be considered in the continuation of the study under the assumption that the series is convergent.

    Lemma 5.8. Let Υω and B=(brs) be an infinite matrix. If B(p(M):Υ), then H(p:Υ) and Bu=Hν hold for all up(M) and 1p+.

    Proof. It is obvious from Lemma 5.5. So, we omit it.

    Theorem 5.9. For 1<p<+:

    (ⅰ) If B(p(M):), then

    0KBχlim supr(s|hrs|q)1q,

    and KB is compact if

    limr(s|hrs|q)1q=0.

    (ⅱ) If B(p(M):c0), then

    KBχ=lim supr(s|hrs|q)1q,

    and KB is compact iff

    limr(s|hrs|q)1q=0.

    (ⅲ) If B(p(M):1), then

    limjB(j)(p(M):1)KBχ4.limjB(j)(p(M):1),

    and KB is compact iff

    limjB(j)(p(M):1)=0,

    where B(j)(p(M):1)=supEFj(s|rEhrs|q)1q for all jN.

    Proof. (ⅰ) Let B(p(M):) and u=(us)p(M). Since the series sbrsus converges for each rN, Br(p(M))β. From Lemma 5.6 (ⅲ), we reach that Brp(M)=(s|hrs|q)1q. In that case, from Lemma 5.7 (ⅰ), we see that

    0KBχlim supr(s|hrs|q)1q,

    and KB is compact if

    limr(s|hrs|q)1q=0.

    (ⅱ) Suppose that B(p(M):c0). Since Brp(M)=(s|hrs|q)1q for each rN and from Lemma 5.7 (ⅱ), we see that

    KBχ=lim supr(s|hrs|q)1q,

    and KB is compact iff

    limr(s|hrs|q)1q=0.

    (ⅲ) Let B(p(M):1). From Lemma 5.6, rEBrp(M)=rEHrq. Thus, from Lemma 5.7 (ⅲ), we see that

    limj(supEFjs|rEhrs|q)1qKBχ4.limj(supEFjs|rEhrs|q)1q,

    and KB is compact iff

    limj(supEFjs|rEhrs|q)1q=0.

    Theorem 5.10. The following statements hold:

    (ⅰ) If B((M):), then

    0KBχlim suprs|hrs|,

    and KB is compact if

    limrs|hrs|=0.

    (ⅱ) If B((M):c0), then

    KBχ=lim suprs|hrs|,

    and KB is compact iff

    limrs|hrs|=0.

    (ⅲ) If B((M):1), then

    limjB(j)((M):1)KBχ4.limjB(j)((M):1),

    and KB is compact iff

    limjB(j)((M):1)=0,

    where B(j)((M):1)=supEFj(s|rEhrs|).

    Proof. This can be proven in the same manner as Theorem 5.9, so we omit the proof here.

    Theorem 5.11. (ⅰ) If B(1(M):), then

    0KBχlim supr(sups|hrs|),

    and KB is compact if

    limr(sups|hrs|)=0.

    (ⅱ) If B(1(M):c0), then

    KBχ=lim supr(sups|hrs|),

    and KB is compact iff

    limr(sups|hrs|)=0.

    Proof. It can be seen this in a similar way to the proof of Theorem 5.9.

    Lemma 5.12. [22] If Λ has AK property or Λ= and B(Λ:c), then

    12lim suprBrbΛKBχlim suprBrbΛ,

    and KB is compact iff

    limrBrbΛ=0,

    where b=(bs) and bs=limrbrs.

    Theorem 5.13. If B(p(M):c) for 1<p<+, then

    12lim supr(s|hrshs|q)1qKBχlim supr(s|hrshs|q)1q,

    and KB is compact iff

    limr(s|hrshs|q)1q=0.

    Proof. Let B(p(M):c). In that case, it is obtained that H(p:c) by Lemma 5.8. By the aid of the Lemma 5.12, we reache that

    12lim suprHrhpKBχlim suprHrhp.

    This implies that, by Lemma 5.6 (ⅲ),

    12lim supr(s|hrshs|q)1qKBχlim supr(s|hrshs|q)1q,

    holds. Hence, we conclude with Lemma 5.12 that KB is compact iff

    limr(s|hrshs|q)1q=0.

    Theorem 5.14. If B((M):c), in this case

    12lim supr(s|hrshs|)KBχlim supr(s|hrshs|),

    and KB is compact iff

    limr(s|hrshs|)=0.

    Proof. It can be seen this in a similar way to the proof of Theorem 5.13.

    Theorem 5.15. If B(1(M):c), then

    12lim supr(sups|hrshs|)KBχlim supr(sups|hrshs|),

    and KB is compact iff

    limr(sups|hrshs|)=0.

    Proof. It can be seen this also in a similar way to the proof of Theorem 5.13.

    In this study, as an example of the application of matrix summability methods to Banach spaces, two new sequence spaces are constructed as the domains of the Motzkin matrix operator defined by Erdem et al. [10] in the sequence spaces p and , some algebraic and topological properties of these spaces are revealed, their duals are calculated, some matrix classes concerning the new spaces are characterized and finally, the compactness criteria of some operators on these spaces are expressed with the help of the Hausdorff measure of non-compactness.

    In our future work, we plan to act with the idea expressed above and obtain new normed and paranormed sequence spaces in this direction.

    Creating infinite matrices with the help of special number sequences and thus obtaining new normed or paranormed sequence spaces and also examining some properties in these spaces (e.g., completeness, inclusion relations, Schauder basis, duals, matrix transformations, compact operators and core theorems) can be suggested as an idea to researchers who want to study in this field.

    Mr : rth Motzkin number
    M : Motzkin matrix
    ω : space of all real sequences
    : space of bounded sequences
    c : space of convergent sequences
    c0 : space of null sequences
    p : space of absolutely p-summable sequences
    bs : space of bounded series
    cs : space of convergent series
    (Λ:Υ) : class of matrix mappings described from Λ to Υ
    ΥB : matrix domain of B in Υ
    ΛΥ : linear isomorphism between the spaces Λ and Υ
    . : floor function
    N : set of positive integers with zero
    F : collection of all finite subsets of N
    Fj : subcollection of F consisting of subsets of N with elements that are greater than j
    Λα : α-dual of the sequence space Λ
    Λβ : β-dual of the sequence space Λ
    Λγ : γ-dual of the sequence space Λ
    DΛ : unit sphere of the normed space Λ
    C(Λ:Υ) : set of all bounded linear mappings from Λ to Υ
    χ(P) : Hausdorff measure of non-compactness of a bounded set P
    Kχ : Hausdorff measure of non-compactness of a linear mapping K
    Q(uj,nj) : open ball centred at uj with radius nj

    The author declares he is not used Artificial Intelligence (AI) tools in the creation of this article.

    The author would like to thank the referees for valuable comments improving the paper.

    The author declares no conflicts of interest.



    [1] M. Aigner, Motzkin numbers, Europ. J. Combinatorics, 19 (1998), 663–675. https://doi.org/10.1006/eujc.1998.0235 doi: 10.1006/eujc.1998.0235
    [2] B. Altay, F. Başar, Certain topological properties and duals of the matrix domain of a triangle matrix in a sequence space, J. Math. Anal. Appl., 336 (2007), 632–645. https://doi.org/10.1016/j.jmaa.2007.03.007 doi: 10.1016/j.jmaa.2007.03.007
    [3] F. Başar, E. Malkowsky, The characterization of compact operators on spaces of strongly summable and bounded sequences, Appl. Math. Comput., 217 (2011), 5199–5207. https://doi.org/10.1016/j.amc.2010.12.007 doi: 10.1016/j.amc.2010.12.007
    [4] F. Başar, Summability theory and its applications, 2nd ed., New York: CRC Press/Taylor & Francis Group, 2022. https://doi.org/10.1201/9781003294153
    [5] J. Boos, Classical and modern methods in summability, Oxford University Press, 2000.
    [6] M. C. Dağlı, A novel conservative matrix arising from Schröder numbers and its properties, Linear Mult. Algebra, 71 (2023), 1338–1351. https://doi.org/10.1080/03081087.2022.2061401 doi: 10.1080/03081087.2022.2061401
    [7] M. C. Dağlı, Matrix mappings and compact operators for Schröder sequence spaces, Turkish J. Math., 46 (2022), 2304–2320. https://doi.org/10.55730/1300-0098.3270 doi: 10.55730/1300-0098.3270
    [8] S. Demiriz, S. Erdem, Mersenne matrix operator and its application in p-Summable sequence space, Commun. Adv. Math. Sci., 7 (2024), 42–55. https://doi.org/10.33434/cams.1414791 doi: 10.33434/cams.1414791
    [9] S. Erdem, S. Demiriz, On the new generalized block difference sequence space, Appl. Appl. Math., 5 (2019), 68–83.
    [10] S. Erdem, S. Demiriz, A. Şahin, Motzkin sequence spaces and motzkin core, Numer. Funct. Anal. Optim., 45 (2024), 283–303. https://doi.org/10.1080/01630563.2024.2333250 doi: 10.1080/01630563.2024.2333250
    [11] M. İlkhan, P. Z. Alp, E. E. Kara, On the spaces of linear operators acting between asymmetric cone normed spaces, Mediterr. J. Math., 15 (2018). https://doi.org/10.1007/s00009-018-1182-0
    [12] M. İlkhan, A new conservative matrix derived by Catalan numbers and its matrix domain in the spaces c and c0, Linear Mult. Algebra, 68 (2019), 417–434. https://doi.org/10.1080/03081087.2019.1635071 doi: 10.1080/03081087.2019.1635071
    [13] M. İlkhan, E. E. Kara, Matrix transformations and compact operators on Catalan sequence spaces, J. Math. Anal. Appl., 498 (2021), 124925. https://doi.org/10.1016/j.jmaa.2021.124925 doi: 10.1016/j.jmaa.2021.124925
    [14] E. E. Kara, M. Başarır, An application of Fibonacci numbers into infinite Toeplitz matrices, Caspian J. Math. Sci., 1 (2012), 43–47.
    [15] E. E. Kara, Some topological and geometrical properties of new Banach sequence spaces, J. Inequal. Appl., 2013. https://doi.org/10.1186/1029-242X-2013-38
    [16] M. Karakaş, M. C. Dağlı, Some topologic and geometric properties of new Catalan sequence spaces, Adv. Oper. Theory, 8 (2023). https://doi.org/10.1007/s43036-022-00243-9
    [17] M. Karakaş, H. Karabudak, An application on the Lucas numbers and infinite Toeplitz matrices, Cumhuriyet Sci. J., 38 (2017), 557–562. https://doi.org/10.17776/csj.340510 doi: 10.17776/csj.340510
    [18] M. Karakaş, A. M. Karakaş, New Banach sequence spaces that is defined by the aid of Lucas numbers, Iğdır Univ. J. Inst. Sci. Tech., 7 (2017), 103–111.
    [19] M. Karakaş, On the sequence spaces involving Bell numbers, Linear Mult. Algebra, 71 (2022), 2298–2309. https://doi.org/10.1080/03081087.2022.2098225 doi: 10.1080/03081087.2022.2098225
    [20] E. Malkowsky, V. Rakocevic, An introduction into the theory of sequence spaces and measure of noncompactness, Zbornik Radov., 9 (2000), 143–234.
    [21] T. Motzkin, Relations between hypersurface cross ratios, and a combinatorial formula for partitions of a polygon, for permanent preponderance, and for nonassociative products, Bull. Amer. Math. Soe., 54 (1948), 352–360. https://doi.org/10.1090/S0002-9904-1948-09002-4 doi: 10.1090/S0002-9904-1948-09002-4
    [22] M. Mursaleen, A. K. Noman, Compactness by the Hausdorff measure of noncompactness, Nonlinear Anal., 73 (2010), 2541–2557. https://doi.org/10.1016/j.na.2010.06.030 doi: 10.1016/j.na.2010.06.030
    [23] M. Mursaleen, A. K. Noman, Applications of the Hausdorffmeasure of noncompactness in some sequence spaces of weighted means, Comput. Math. Appl., 60 (2010), 1245–1258. https://doi.org/10.1016/j.camwa.2010.06.005 doi: 10.1016/j.camwa.2010.06.005
    [24] M. Mursaleen, F. Başar, Sequence spaces: Topic in modern summability theory, CRC Press, 2020. https://doi.org/10.1201/9781003015116
    [25] V. Rakocevic, Measures of noncompactness and some applications, Filomat, 12 (1998), 87–120.
    [26] M. Stieglitz, H. Tietz, Matrix transformationen von folgenraumen eine ergebnisbersicht, Math Z., 154 (1977), 1–16. https://doi.org/10.1007/BF01215107 doi: 10.1007/BF01215107
    [27] A. Wilansky, Summability through functional analysis, Elsevier, 1984.
    [28] T. Yaying, B. Hazarika, S. A. Mohiuddine, Domain of Padovan q-difference matrix in sequence spaces p and , Filomat, 36 (2022), 905–919. https://doi.org/10.2298/FIL2203905Y doi: 10.2298/FIL2203905Y
    [29] T. Yaying, B. Hazarika, O. M. K. S. K. Mohamed, A. A. Bakery, On new Banach sequence spaces involving Leonardo numbers and the associated mapping ideal, J. Funct. Spaces, 2022 (2022), 1–21. https://doi.org/10.1155/2022/8269000 doi: 10.1155/2022/8269000
  • This article has been cited by:

    1. Sezer Erdem, Schröder–Catalan Matrix and Compactness of Matrix Operators on Its Associated Sequence Spaces, 2024, 16, 2073-8994, 1317, 10.3390/sym16101317
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(794) PDF downloads(46) Cited by(1)

Figures and Tables

Tables(3)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog