Psoriasis after vaccination, namely “Psoriasis Vaccinalis”, has been described with several vaccines. More recently, emerging reports have linked coronavirus (COVID-19) vaccines to exacerbation or new-onset psoriasis. We report 2 female patients who developed severe new-onset palmoplantar pustular psoriasis following vaccination against COVID-19 (one received a deactivated vaccine, and one received an mRNA vaccine). The patients developed the psoriatic lesions 1 and 2 weeks after the vaccinations, respectively. The patients were primarily diagnosed in a primary care facility or through self-diagnosis with skin/deep soft tissue infections of the hand or foot, and started topical antibiotics before seeking dermatological consultations, where the diagnosis of psoriasis was concluded and confirmed by skin biopsies. The patients were successfully treated with anti-psoriasis topical and/or systemic medications. Psoriasis Vaccinalis has been previously described with several vaccines. COVID-19 vaccines are not an exception, and new-onset pustular psoriasis can occur after any type of COVID-19 vaccination, thereby mimicking a skin infection in its early stage. Primary care clinicians should remain vigilant for potential cases of vaccine-induced psoriasis to prevent misdiagnosis and facilitate a timely referral to dermatology.
Citation: Raghda S. El Mashaari, Ameera S. Al Ali, Zaidoon M. Abdel Hadi, Mahmoud Talaat Sabri, Ahmed Nageeb Mahmoud. COVID-19 vaccine-induced new-onset pustular psoriasis mimicking skin infection of the hand and foot: A report of two cases[J]. AIMS Allergy and Immunology, 2025, 9(2): 136-143. doi: 10.3934/Allergy.2025010
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Abstract
Psoriasis after vaccination, namely “Psoriasis Vaccinalis”, has been described with several vaccines. More recently, emerging reports have linked coronavirus (COVID-19) vaccines to exacerbation or new-onset psoriasis. We report 2 female patients who developed severe new-onset palmoplantar pustular psoriasis following vaccination against COVID-19 (one received a deactivated vaccine, and one received an mRNA vaccine). The patients developed the psoriatic lesions 1 and 2 weeks after the vaccinations, respectively. The patients were primarily diagnosed in a primary care facility or through self-diagnosis with skin/deep soft tissue infections of the hand or foot, and started topical antibiotics before seeking dermatological consultations, where the diagnosis of psoriasis was concluded and confirmed by skin biopsies. The patients were successfully treated with anti-psoriasis topical and/or systemic medications. Psoriasis Vaccinalis has been previously described with several vaccines. COVID-19 vaccines are not an exception, and new-onset pustular psoriasis can occur after any type of COVID-19 vaccination, thereby mimicking a skin infection in its early stage. Primary care clinicians should remain vigilant for potential cases of vaccine-induced psoriasis to prevent misdiagnosis and facilitate a timely referral to dermatology.
1.
Introduction
All real sequence's space is expressed by ω. Any Λ⊂ω is called a sequence space. Frequently encountered sequence spaces can be denoted as ℓ∞ (space of bounded sequences), c (space of convergent sequences), c0 (space of null sequences), and ℓp(1≤p<∞) (space of absolutely p-summable sequences). These are Banach spaces with ‖u‖ℓ∞=‖u‖c=‖u‖c0=supr∈N|ur| and ‖u‖ℓp=(∑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)∈Λ⊂ω can be expressed uniquely as u=∑rure(r), then the space Λ satisfies the AK-property.
Consider the multiplier set (Λ⋇Υ) defined by
(Λ⋇Υ)={τ=(τr)∈ω:τu=(τrur)∈Υ for allu∈Λ},
for Λ,Υ⊂ω. Thus, the α-, β- and γ-duals of Λ are expressed by
Λα=(Λ⋇ℓ1),Λβ=(Λ⋇cs) andΛγ=(Λ⋇bs).
Br refers to the rth row of an infinite matrix B=(brs) with real terms. Additionally, if the series is convergent for all r∈N, (Bu)r=∑sbrsus is a B-transform of u∈ω. The infinite matrix B is a matrix mapping from Λ to Υ, if Bu∈Υ for all u∈Λ. The class of all matrix mappings described from Λ to Υ is given by (Λ:Υ). Additionally, B∈(Λ:Υ) iff Br∈Λβ and Bu∈Υ for all u∈Λ. The set
ΥB={u∈ω:Bu∈Υ},
(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 Λ≅Υ.
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 (r∈N) have various applications in geometry, combinatorics, and number theory, and they are represented by the following sequence:
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,s∈N constructed with the help of Motzkin numbers and relation (1.2) by Erdem et al. [10] can be written follows:
mrs:={MsMr−sMr+2−Mr+1, if 0≤s≤r,0, if s>r.
(1.3)
It is possible to state the Motzkin matrix more clearly as follows:
Furthermore, it is also known from [10] that the Motzkin matrix M is conservative, that is, M∈(c:c) and the inverse M−1=(m−1rs) of the Motzkin matrix M is
m−1rs:={(−1)r−sMs+2−Ms+1Mrπr−s, if 0≤s≤r,0, if s>r,
for all r∈N∖{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+2−Mr+1r∑s=0MsMr−sus,(r∈N).
(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 ℓp(1≤p<+∞) and ℓ∞. 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.
2.
Motzkin sequence spaces ℓp(M) and ℓ∞(M)
In this part, we introduce the BK-sequence spaces ℓp(M) and ℓ∞(M) with the help of the Motzkin matrix which are linearly isomorphic to ℓp and ℓ∞, respectively. Finally, the Schauder basis of ℓp(M) and the inclusion relations of the spaces are presented.
Now, we may present the Motzkin sequence spaces ℓp(M) and ℓ∞(M) as follows:
for 1≤p<+∞. Then, ℓp(M) and ℓ∞(M) can be rewritten as ℓp(M)=(ℓp)M and ℓ∞(M)=(ℓ∞)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.ℓp(M) and ℓ∞(M) are BK-spaces with
‖u‖ℓp(M)=(∑r|1Mr+2−Mr+1r∑s=0MsMr−sus|p)1p,
and
‖u‖ℓ∞(M)=supr∈N|1Mr+2−Mr+1r∑s=0MsMr−sus|,
respectively.
Proof. In the proof of this theorem, it will be used that ℓp and ℓ∞ are BK-spaces. It is known from Wilansky [27] that ΥB is BK-space with ‖u‖ΥB=‖Bu‖Υ if B is triangle and Υ is a BK-space. Consequently, ℓp(M) and ℓ∞(M) are BK-spaces with the norms ‖.‖ℓp(M) and ‖.‖ℓ∞(M), respectively. □
Theorem 2.2.ℓp(M)≅ℓp and ℓ∞(M)≅ℓ∞.
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:ℓp(M)→ℓp, K(u)=Mu is linear and since K(u)=0⇒u=0, K is injective. Consider the sequences ν=(νs)∈ℓp and u=(us)∈ω with
for ∑r−is=0(−1)sMr−s−iπs=0, if r≠i, then K is surjective. Additionally, since the relation ‖u‖ℓp(M)=‖Mu‖ℓp 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 u∈Λ, there is a unique sequence (σr) of scalars such that
‖u−r∑s=0σsηs‖⟶0,
as r→+∞. This can be stated as u=∑sσsηs.
The inverse image of the basis (e(s))s∈N of ℓp becomes the basis of ℓp(M)(1≤p<+∞) since K:ℓp(M)→ℓp 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)∈ℓp(M) expressed by
η(s)r:={(−1)r−sMs+2−Ms+1Mrπr−s,if0≤s≤r,0,ifs>r,
is a Schauder basis for ℓp(M) and the unique representation of any u∈ℓp(M) is stated as u=∑sσsη(s) for 1≤p<+∞ and σs=(Mu)s.
It is worth noting that since every normed linear space with a Schauder basis is separable, the sequence space ℓp(M) is seperable.
Theorem 2.4.The inclusion ℓp(M)⊂ℓ˜p(M) strictly holds for 1≤p<˜p<+∞.
Proof. For the first part, it is sufficient to show that every element taken from ℓp(M) is in ℓ˜p(M). Let us take u=(us)∈ℓp(M) such that Mu∈ℓp. Furthermore, it is known that ℓp⊂ℓ˜p for 1≤p<˜p<+∞. Then, Mu∈ℓ˜p and u=(us)∈ℓ˜p(M).
If ˜ν=M˜u∈ℓ˜p∖ℓp, then the rest of the proof is completed. □
Theorem 2.5.ℓ∞⊂ℓ∞(M).
Proof. For the proof, it is necessary to show that every element taken from ℓ∞ is in ℓ∞(M). For u=(us)∈ℓ∞, it is clear that
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,
ℓξ1=ℓ∞,ℓξ∞=cξ0=cξ=ℓ1 and ℓξp=ℓq, where 1<p,q<+∞ with q=pp−1.
Additionally, for any Λ∈ω, Λα⊂Λβ⊂Λγ. If (brs)s∈N∈Λβ for all r∈N, 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
sups∈N∑r|brs|<∞,
(3.1)
sups∈N∑r|brs|p<∞,
(3.2)
supr,s∈N|brs|<∞,
(3.3)
limr→+∞brs exists for all s∈N,
(3.4)
limr→+∞brs=0,
(3.5)
supE∈F∑s|∑r∈Ebrs|q<∞,
(3.6)
supr∈N∑s|brs|q<∞,
(3.7)
supE∈F∑r|∑s∈Ebrs|<∞,
(3.8)
supE∈F∑r|∑s∈Ebrs|p<∞,
(3.9)
supr∈N∑s|brs|<∞,
(3.10)
limr→+∞∑s|brs|=∑s|limr→+∞brs|,
(3.11)
limr→+∞∑s|brs|=0,
(3.12)
where, the F⊂N 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 ϖ1−ϖ7 which will be used to compute the duals:
ϖ1={τ=(τs)∈ω:supE∈F∑s|∑r∈E(−1)r−sMs+2−Ms+1Mrπr−sτr|q<∞},ϖ2={τ=(τs)∈ω:sups∈N∑r|(−1)r−sMs+2−Ms+1Mrπr−sτr|<∞},ϖ3={τ=(τs)∈ω:supE∈F∑r|∑s∈E(−1)r−sMs+2−Ms+1Mrπr−sτr|<∞},ϖ4={τ=(τs)∈ω:limr→+∞r∑i=s(−1)i−sMs+2−Ms+1Miπi−sτi exists for each s∈N},ϖ5={τ=(τs)∈ω:supr∈N∑s|r∑i=s(−1)i−sMs+2−Ms+1Miπi−sτi|q<∞},ϖ6={τ=(τs)∈ω:supr,s∈N|r∑i=s(−1)i−sMs+2−Ms+1Miπi−sτi|<∞},ϖ7={τ=(τs)∈ω:limr→+∞∑s|r∑i=s(−1)i−sMs+2−Ms+1Miπi−sτi|=∑s|∞∑i=s(−1)i−sMs+2−Ms+1Miπi−sτi|}.
The proofs of (ⅱ) and (ⅲ) can be seen similarly to the first part through the aid of the conditions of the classes (ℓ1:ℓ1) and (ℓ∞:ℓ1), respectively, from Table 1. Therefore, they are ommitted. □
Theorem 3.2.The following statements hold:
(ⅰ)(ℓp(M))β=ϖ4∩ϖ5, (1<p<+∞),
(ⅱ)(ℓ1(M))β=ϖ4∩ϖ6,
(ⅲ)(ℓ∞(M)β=ϖ4∩ϖ7.
Proof.(ⅰ) Consider that τ=(τs)∈ω and u∈ℓp(M) with ν∈ℓp as (1.5). By considering the Eq (2.1), we can see that
for every r,s∈N. Then, from (3.14), τu∈cs while u=(us)∈ℓp(M) iff ψ=(ψr)∈c while ν∈ℓp. In this case, τ∈(ℓp(M))β iff O∈(ℓp:c). Consequently, from the conditions of (ℓp: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 (ℓ1:c) and (ℓ∞:c), respectively, from the Table 1. Therefore, they are ommitted too.
Theorem 3.3.The following statements hold:
(ⅰ)(ℓp(M))γ=ϖ5,(1<p<+∞),
(ⅱ)(ℓ1(M))γ=ϖ6,
(ⅲ)(ℓ∞(M)γ=ϖ5 with q=1.
Proof. This can be done similarly with Theorem 3.2 by considering with together the classes (ℓp:ℓ∞), (ℓ1:ℓ∞), and (ℓ∞:ℓ∞) from Table 1 with O=(ors) expressed by (3.15). □
4.
Matrix mappings
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,Υ∈ω, infinite matrices H(r)=(h(r)is) and H∗=(h∗rs) described as
for all i,r∈N. Since Bu exists, then H(r)∈(Λ:c). By passing limit for i→+∞ in the relation (4.3), Bu=H∗ν. Since Bu∈Υ, H∗ν∈Υ and so H∗∈(Λ:Ψ).
Conversely, let us suppose that H(r)∈(Λ:c) and H∗∈(Λ:Υ). Then, h∗rs∈Λβ which gives us (brs)s∈N∈(Λ(M))β for all r∈N. Hence, Bu exists for all u∈Λ(M). Therefore, from relation (4.3) for i→+∞, Bu=H∗ν. Thus B∈(Λ(M):Υ), which is desired result. □
Corollary 4.2.Consider the infinite matrices H(r)=(h(r)is) and H∗=(h∗rs) 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∈{ℓ1,ℓp,ℓ∞}, Υ∈{ℓ1,ℓp,ℓ∞,c,c0} and 1<p<+∞.
Table 2.
Characterizations of the classes (Λ(M):Υ), where Λ∈{ℓ1,ℓp,ℓ∞}, Υ∈{ℓ1,ℓp,ℓ∞,c,c0} and 1<p<+∞.
(Λ(M)↓:Υ→)
ℓ1
ℓp
ℓ∞
c
c0
ℓ1(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)∗
ℓp(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)∗
ℓ∞(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 3.1≤λ≤3.12.
Theorem 4.3.Consider the infinite matrices ˜B=(˜brs) and B=(brs) described with the relation
˜brs=r∑j=0MjMr−jMr+2−Mr+1bjs.
(4.4)
In that case, B∈(Λ:Υ(M)) iff ˜B∈(Λ:Υ) for Λ∈{ℓ1,ℓp,ℓ∞,c,c0} and Υ∈{ℓ1,ℓp,ℓ∞}.
Proof. Consider that the infinite matrices ˜B and B described with the relation (4.4), Λ∈{ℓ1,ℓp,ℓ∞,c,c0}, and Υ∈{ℓ1,ℓp,ℓ∞}. For any u=(us)∈Λ,
∞∑s=0˜brsus=r∑j=0MjMr−jMr+2−Mr+1∞∑s=0bjsus.
This means that ˜Br(u)=Mr(Bu) for all r∈N, which implies that Bu∈Υ(M) iff ˜Bu∈Υ for every u∈Λ. 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∈{ℓ1,ℓp,ℓ∞,c,c0} and Υ∈{ℓ1,ℓp,ℓ∞}.
Table 3.
Characterizations of (Λ:Υ(M)), where Λ∈{ℓ1,ℓp,ℓ∞,c,c0} and Υ∈{ℓ1,ℓp,ℓ∞}.
5.
Compactness by Hausdorff measure of non-compactness
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 ‖u‖⋄Λ is expressed by
‖u‖⋄Λ=supx∈DΛ|∑susxs|,
for a BK-space Λ⊃Ω and u=(us)∈ω for all finite sequences' space, Ω, provided that the series is finite and u∈Λβ.
The set of all bounded linear mappings from Λ to Υ is denoted by C(Λ:Υ).
Lemma 5.2.[20] There exists KB∈C(Λ:Υ) as KB(u)=Bu for BK-spaces Λ and Υ, as well as for all u∈Λ and B∈(Λ:Υ).
Lemma 5.3.[20] If B∈(Λ:Υ), then ‖KB‖=‖B‖(Λ:Υ)=supr∈N‖Br‖⋄Λ<∞ for Υ∈{c0,c,ℓ∞} and the BK-space Λ⊃Ω.
The Hausdorff measure of non-compactness of a bounded set P in the metric space Λ is stated with
χ(P)=inf{ϵ>0:P⊂∪rj=1Q(uj,nj),uj∈Λ,nj<ϵ,r∈N∖0},
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 P⊂ℓp is bounded and mapping Ψn:ℓp⟶ℓp stated by Ψn(u)=(u0,u1,u2,...,un,0,0,...) for all u=(us)∈ℓp, 1≤p<+∞ and n∈N. Then,
χ(P)=limn→∞(supu∈P‖(I−Ψn)(u)‖ℓp),
for the identity operator I on ℓp.
A linear mapping K is compact if (K(u)) has a convergent subsequence in Υ for all u=(us)∈ℓ∞∩Λ for the Banach spaces Λ and Υ.
The Hausdorff measure of non-compactness ‖K‖χ of K is expressed with ‖K‖χ=χ(K(DΛ)). In that case, K is compact iff ‖K‖χ=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)j−sMs+2−Ms+1Mjπj−sxj,
(5.1)
for all s∈N.
Lemma 5.5.Let us consider the sequence x=(xs)∈(ℓp(M))β for 1≤p≤∞. In that case, y=(ys)∈ℓq and
∑sxsus=∑sysνs,
(5.2)
for all u=(us)∈ℓp(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‖⋄ℓ∞(M)=∑s|ys|<∞ for all x=(xs)∈(ℓ∞(M))β.
(ⅱ)‖x‖⋄ℓ1(M)=sups|ys|<∞ for all x=(xs)∈(ℓ1(M))β.
(ⅲ)‖x‖⋄ℓp(M)=(∑s|ys|q)1q<∞ for all x=(xs)∈(ℓp(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)∈ℓ1 and (5.2) holds for x=(xs)∈(ℓ∞(M))β and for all u=(us)∈ℓ∞(M). Since ‖u‖ℓ∞(M)=‖ν‖ℓ∞ with (1.5), then u∈Dℓ∞(M) if and only if ν∈Dℓ∞. Thus, we can write the equality ‖x‖⋄ℓ∞(M)=supu∈Dℓ∞(M)|∑sxsus|=supν∈Dℓ∞|∑sysνs|=‖y‖⋄ℓ∞. By the aid of the Lemma 5.1, it follows that ‖x‖⋄ℓ∞(M)=‖y‖⋄ℓ∞=‖y‖ℓ1=∑s|ys|<∞. □
and KB is compact iff limj(supE∈Fj‖∑r∈EBr‖⋄Λ)=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∗=(h∗rs) 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 Υ⊂ω and B=(brs) be an infinite matrix. If B∈(ℓp(M):Υ), then H∗∈(ℓp:Υ) and Bu=H∗ν hold for all u∈ℓp(M) and 1≤p≤+∞.
Proof. It is obvious from Lemma 5.5. So, we omit it. □
where ‖B‖(j)(ℓp(M):ℓ1)=supE∈Fj(∑s|∑r∈Eh∗rs|q)1q for all j∈N.
Proof.(ⅰ) Let B∈(ℓp(M):ℓ∞) and u=(us)∈ℓp(M). Since the series ∑sbrsus converges for each r∈N, Br∈(ℓp(M))β. From Lemma 5.6 (ⅲ), we reach that ‖Br‖⋄ℓp(M)=(∑s|h∗rs|q)1q. In that case, from Lemma 5.7 (ⅰ), we see that
0≤‖KB‖χ≤lim supr(∑s|h∗rs|q)1q,
and KB is compact if
limr(∑s|h∗rs|q)1q=0.
(ⅱ) Suppose that B∈(ℓp(M):c0). Since ‖Br‖⋄ℓp(M)=(∑s|h∗rs|q)1q for each r∈N and from Lemma 5.7 (ⅱ), we see that
‖KB‖χ=lim supr(∑s|h∗rs|q)1q,
and KB is compact iff
limr(∑s|h∗rs|q)1q=0.
(ⅲ) Let B∈(ℓp(M):ℓ1). From Lemma 5.6, ‖∑r∈EBr‖⋄ℓp(M)=‖∑r∈EH∗r‖⋄ℓq. Thus, from Lemma 5.7 (ⅲ), we see that
Proof. It can be seen this also in a similar way to the proof of Theorem 5.13. □
6.
Conclusions
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 ℓp and ℓ∞, 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.
List of abbreviations and symbols
Mr
: rth Motzkin number
M
: Motzkin matrix
ω
: space of all real sequences
ℓ∞
: space of bounded sequences
c
: space of convergent sequences
c0
: space of null sequences
ℓp
: 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
Λ≅Υ
: 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
‖K‖χ
: Hausdorff measure of non-compactness of a linear mapping K
Q(uj,nj)
: open ball centred at uj with radius nj
Use of AI tools declaration
The author declares he is not used Artificial Intelligence (AI) tools in the creation of this article.
Acknowledgments
The author would like to thank the referees for valuable comments improving the paper.
Conflict of interest
The author declares no conflicts of interest.
Conflicts of interest
All authors declare that there are no competing interests.
Author contributions
Raghda Al Mashaari: Investigations, Methodology, Data Collection, Formal analysis; Ameera Al Ali: Investigations, Methodology, Data Collection, Formal analysis, Manuscript writing–original draft; Zaidoon Abdel Hadi: Investigations, Methodology, Data Collection, Formal analysis; Mahmoud Talaat Sabri: Data Curation, Conceptualization, Methodology; Ahmed Nageeb Mahmoud: Conceptualization, Data Curation, Methodology, Manuscript writing–original draft, Manuscript writing–review and editing.
Ethics approval of research
Informed consent has been obtained from the patients for the anonymous use of the clinical data for research purposes.
Institutional Review Board approval was waived due to the nature of the research (case report, two cases); however, the authors declare that the research complies with HIPAA.
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Note: The symbol "∙" represents the conditions of the classes that are unknown or not interest to this study.
(Λ(M)↓:Υ→)
ℓ1
ℓp
ℓ∞
c
c0
ℓ1(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)∗
ℓp(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)∗
ℓ∞(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 3.1≤λ≤3.12.
(Λ↓:Υ(M))→)
ℓ1(M)
ℓp(M)
ℓ∞(M)
ℓ1
(3.1)
(3.2)
(3.3)
ℓp
(3.6)
∙
(3.7)
ℓ∞
(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).
Figure 1. A, B: Localized pinkish erythematous patch studded with a few yellowish pustules and hemorrhagic blisters on the heel and the plantar aspect of the big toe. C–J: Serial clinical photos showing gradual response to Guselkumab therapy, which eventually resulted in a resolution of the skin lesions
Figure 2. Skin biopsy showing keratotic epidermis with psoriasiform blunted rete ridges with a surface parakeratosis. Numerous neutrophils infiltrated into the epidermis and collected subcorneally. The papillary dermis showed mixed perivascular inflammation with scattered neutrophils
Figure 3. A, B: Localized pinkish erythematous patch studded with a few yellowish pustules on the palm of the right hand. B, C: Serial clinical photos showing the response to the topical anti-psoriasis therapy. D, E: Punch skin biopsy showing a papillary dermis and neutrophilic infiltration into the epidermis