Introduction

Sequence spaces have played a vital role across various branches of mathematics, such as functional analysis, operator theory, and approximation theory. The importance of sequence spaces has sparked considerable interest among researchers in summability theory. They have introduced and investigated different types of sequence spaces to uncover their unique properties. For example denote the spaces of all bounded, p-summable, convergent and null sequences, respectively. Further, cs,cs₀,bs denote the spaces of all convergent, null and bounded series, respectively. Throughout the paper, we will denote as the set of all natural, real and complex numbers, respectively. A sequence space X is called an FK-space if it is a complete linear metric space with continuous coordinates and for all , and a normed FK-space is called a BK-space. For example, c,c₀ and are BK spaces with the norm . Also, l_p is a BK-space with the norm defined by

Let denote the set of all real sequences and the set of all finite sequences, respectively. An FK-space is said to have AK if every has a unique representation where is the sequence whose only non-zero term is 1 in the place for each [1]. The spaces c₀ and l_p have AK [2].

The primary objective of classical theory revolves around the generalization of convergence concepts for both series and sequences. Its main goal is to provide a framework through which limits can be assigned to series and sequences that do not exhibit convergence. This is achieved through the use of transformations defined by infinite matrices. The preference for utilizing matrices, rather than general linear mappings, is based on the fact that a linear mapping between two sequence spaces can be represented by an infinite matrix.

Let Z and W be two sequence spaces and be an infinite matrix of real or complex numbers a_nk, for . Then defines a matrix mapping from Z to W, if , for every sequence , where

(1)

The set of all these matrices is represented by the notation (Z,W). A sequence (x_k) is summable to L if the sequence converges to L. We say that maps Z regularly into W if and we denote the space of such matrices by (Z,W)_reg.

The matrix domain of matrix in the space X is a sequence space which is defined by

It holds considerable importance in the development of new sequence spaces. Moreover, if is a triangle and X is a BK-space then is a BK-space. Many researchers have used this idea to construct new Banach sequence spaces by applying it to special triangles. To know about these sequence spaces, one can see [3–9].

Quantum calculus, often denoted as q-calculus, is a crucial mathematical tool that goes beyond traditional calculus. It plays a transformative role at the intersection of mathematics and physics. For the first time, relations between these topics especially quantum calculus (q-calculus) and q-differential operators were studied by Jackson in [10]. It has a lot of applications in different mathematical areas such as: orthogonal polynomials, hyper-geometric functions, number theory, complex analysis, combinatorics, matrix summability, approximation theory, quantum physics, particle physics, the theory of relativity, etc. For instance, Demiriz and Şahin [11] and Yaying et al. [12] developed sequence spaces using q-Cesàro matrix. Additionally, Yaying et al. [13] studied (p,q) analogue of Euler sequence spaces. Recently, Atabey et al. [14] developed q-Fibonnaci sequence spaces, Ellidokuzoglu and Demiriz [15] constructed q-difference sequence spaces of order m. For more work on q-sequence spaces, one can see [16–20] and references therein. One can see the basic notations on q-calculus in [21]. For q>0 and any positive integer a, a q-number is defined by

For the integers q–binomial coefficients are defined by

where the q–factorial [a]_q! of a is given by

The q-difference operator of is defined by

and the generalized q-difference operator is defined by

The generalized q-difference matrix is given by

and its inverse is given by

Motzkin numbers, named after Theodore Motzkin, are a remarkable sequence of integers. In mathematics, the Motzkin number represents the count of distinct chords that can be drawn between r points on a circle without intersecting. It is important to note that the chords do not necessarily have to touch all the points on the circle.

Motzkin numbers, denoted as , find diverse applications in various mathematical fields such as geometry, combinatorics, and number theory. They possess a recursive nature and hold significant combinatorial properties, which make them valuable tools in multiple areas of mathematics, algorithmic analysis, and even practical applications like coding theory. The Motzkin numbers have proven to be a rich source of mathematical exploration and have contributed to the understanding of fundamental concepts in different disciplines. They are represented by the following sequence:

The Motzkin numbers satisfy the recurrence relations

Another relation provided by the Motzkin numbers is given below:

For more detail on Motzkin numbers one can refer to [22]. Erdem et al. [23] defined the Motzkin matrix as

for all and its inverse is given as

where P₀ = 1 and

for all .

Motivated by the aforementioned works on q-calculus, the application of q-difference operators, and Motzkin numbers in various mathematical and scientific disciplines, in section 2, we construct generalized Motzkin sequence spaces , , , and using q-difference operators. Section 3 explores some topological properties and establishes the Köthe duals of the spaces , , , and . Section 4 presents theorems and corollaries related to matrix transformations from the spaces and into the classical sequence spaces . Section 5 investigates the compactness of certain operators defined on the space . Finally, Section 6 summarizes the main findings of the manuscript.

Some new sequences spaces

We proceed by introducing q-Motzkin matrix as follows:

for all .

Now, using the generalized q-difference matrix and q-Motzkin matrix, we define the generalized q-difference Motzkin sequence spaces , , and as follows:

and

Let u = (u_r) be the transform of a sequence , which is given by the expression:

(2)

for all Define P₀(q) = 1 and

for all .

Then, using Eq (2), we have

(3)

for each Throughout the paper, u and v are related by Eq (2), or equivalently by Eq (3).

Theorem 2.1.

1. 1. , and are BK-spaces endowed with the norm defined by
2. 2. is a BK-spaces endowed with the norm defined by

Proof: The sequence spaces are BK–spaces with their natural norms and is a triangle matrix. Thus, (i) and (ii) follows immediately by using Wilansky’s work [24]. □

Theorem 2.2. The spaces , , and are linearly isomorphic to c, c₀, l_p and respectively.

Proof: We only prove this Theorem for the space and c₀. Define the mapping by for all is invertible which implies that S is a norm preserving linear bijection. Hence, □

Definition 2.1. A sequence is called a Schauder basis for a normed space , if for every , there is a unique sequence of scalars such that

Now, we construct bases for the spaces and . We recall that the matrix domain has a basis if and only if X has a basis. This statement together with Theorem 2.2 gives us the following result:

Theorem 2.3. Let , for all . For every fixed define the sequence of the elements of the space by

Then

1. 1. the set forms the basis for the space and every has a unique representation
2. 2. the set forms the basis for the space and every has a unique representation of the form where

Proof: 1. Clearly, where (e_s) is the sequence with 1 in the place and zeros elsewhere for each Now for and we define

(4)

By applying to Eq (4), we have

Also,

Let be arbitrary. We choose such that

Then, we have

This implies

To show the uniqueness of this representation, let us assume that there exists By the continuity of S transformation defined in the proof of the Theorem 2.2, we get

which is a contradiction with the assumption that for each Hence, the representation is unique.

2. In a similar manner as in (i), one can easily prove (ii). □

Theorem 2.4. .

Proof: Let . Then

Thus, □

α-, β- and γ-duals

In this section, we compute α-, β- and γ-duals of the spaces , , and . Before proceeding, we recall the definitions of α-, β- and γ-duals.

Definition 3.2. The α-, β- and γ-duals of a subset are defined by

respectively.

Before proceeding further, we recall certain lemmas from [25] that are necessary for determining the duals. Throughout the paper, let denotes the family of all finite subsets of and be the compliment of p, that is,

Lemma 3.1. if and only if

(5)

Lemma 3.2. if and only if

(6)

(7)

Lemma 3.3. if and only if Eq (6) holds.

Theorem 3.5. Define the set a₁(q) by

Then .

Proof: Consider the following equality

(8)

for all , where the sequence (u_t) is the -transform of a sequence and the matrix is defined by

From equation Eq (8), we realize that , whenever if and only if whenever Thus, we deduce that z = (z_s) belongs to the α-dual of the spaces if and only if the matrix Thus, from Lemma 3.1, we conclude that the α-dual of the space is a₁(q). □

Theorem 3.6. Define the sets and a₄(q) by

Then and

Proof: Consider the following equality

(9)

for each where the sequence u = (u_t) is the -transform of a sequence and the matrix is defined by

for all From equation Eq (9), we realize that , whenever if and only if whenever Thus, we deduce that z = (z_s) belongs to the β-dual of the space if and only if the matrix Thus, from Lemma 3.2, we have and Thus, . □

Theorem 3.7. The γ-dual of the spaces and is a₃(q).

Proof: The proof is similar to the Theorem 3.6 except that Lemma 3.3 is employed instead of Lemma 3.2. □

Lemma 3.4. [25–27]. The following statements holds true:

1. 1. iff (10)
2. 2. iff (11)(12)
3. 3. iff (13)
4. 4. iff (14)(15)
5. 5.
   1. (a) For iff Eq (11) and Eq (14) hold.
   2. (b) For iff Eq (11) and Eq (15) hold.
6. 6. iff (16) (17)

Theorem 3.8. Define the sets and b₃(q) by

and

Then

1. 1.
2. 2.

Proof: From equation Eq (8), we realize that , whenever if and only if whenever Thus, we deduce that z = (z_p) belongs to the α-dual of the spaces if and only if the matrix Thus, from Lemma 3.4/(vi), we conclude

In a similar manner by utilizing Lemma 3.4/(iii) instead of Lemma 3.4/(vi), we get . Hence, the result. □

Theorem 3.9. Define the sets by

and

Then

1. 1.
2. 2.

Proof: From equation Eq (9), we realize that whenever if and only if whenever This yields that z = (z_s) belongs to the β-dual of the space if and only if the matrix Thus, from Lemma 3.4/(v), we have

In a similar manner by utilizing Lemma 3.4/(ii) instead of Lemma 3.4/(v), we get Hence, the result. □

Theorem 3.10. The following statements hold true:

1. 1.
2. 2. with

Matrix transformations on the spaces and

In this section, we determine necessary and sufficient condition for a matrix transformation from the spaces and to the spaces . The following theorem is fundamental in our investigation.

Theorem 4.11. Let be an arbitrary subset of Then

1. 1. and
2. 2. and

where

and

(18)

for all

Proof: The proof is similar to the proof of Theorem 4.1 of [7] and hence is omitted. □

Now, by using the results presented in [25] together with Theorem 4.11, we obtain the following results.

Corollary 4.1. The following statements hold:

1. 1. iff(19)(20)(21)
   also hold.
2. 2. iff Eq (19) and Eq (20) hold, and(22)(23)
   also hold.
3. 3. ff Eq (19) and Eq (20) hold, and Eq (21) and(24)
   also hold.
4. 4. iff Eq (19) and Eq (20) hold, and(25)
   also hold.
5. 5. iff Eq (19) and Eq (20) hold, and(26)
   also hold.
6. 6. iff Eq (19) and Eq (20) hold, and Eq (26) and(27)
   also hold.
7. 7. iff Eq (19) and Eq (20) hold, and Eq (26) and(28)
   also hold.

Corollary 4.2. The following statements hold:

1. 1. iff Eq (19) and Eq (20) hold, and(29)
   hold and Eq (22) also hold.
2. 2. iff Eq (19), Eq (20) and Eq (29) hold, and Eq (21), Eq (23) and(30)
   also hold.
3. 3. iff Eq (19), Eq (20) and Eq (29) hold, and Eq (21), Eq (24) and(31)
   also hold.
4. 4. iff Eq (19), Eq (20) and Eq (29) hold, and Eq (25) also hold.
5. 5. iff Eq (19), Eq (20) and Eq (29) hold, and Eq (26) also hold.
6. 6. iff Eq (19), Eq (20) and Eq (29) hold, and Eq (26), Eq (27) and(32)
   also hold.
7. 7. iff Eq (19), Eq (20) and Eq (29) hold, and Eq (26), Eq (27) and(33)
   also hold.

Compact operators on the spaces

Let X and Y be Banach spaces. Let U_X denotes the open ball in the space X and be the set of all bounded linear operators , is a Banach space with the norm given by . Further, we denote provided the expression on the right hand side exists and is finite [28]. A linear operator is said to be compact if the domain of T is all of X and for every bounded sequence the sequence (T(x_n)) has a subsequence which converges in Y.

Let M be a bounded set in a metric space X. Then the Hausdorff measure of non-compactness() of M is defined by

where is the open ball centered at z_k and radius r_k for each The operator T is compact if and only if where denotes the Hmnc of T and is defined by For more details about Hmnc, one can see [29] and references therein. The Hmnc of a linear operator plays a role to characterize the compactness of an operator between BK spaces. For the relevant literature, see [30–32].

Let X and Y be any two BK spaces, then every matrix defines a linear operator where for all (see Theorem 3.2.4 of [33]). Moreover, if is a BK-space then (see Theorem 1.23 of [29]).

Let and define a sequence as

for all

Lemma 5.5. [34] and for

Lemma 5.6. [29] Let be the operator defined by for all . Then for any bounded subset M in c₀, we have

where I is the identity operator on c₀.

Lemma 5.7. Let Then and

(34)

for all

Lemma 5.8. for all

Proof: Let Then, by Lemma 5.7, we have and Eq (34) holds. Since, holds, we get if and only if Hence, we conclude that

From Lemma 5.5, it follows that □

Lemma 5.9. Let and L = (l_st) be an infinite matrix. If , then and Lx = Ay for all where L and A are related by the relation Eq (18).

Proof: The proof of this Lemma follows from Lemma 5.7. □

Lemma 5.10. holds for where

Lemma 5.11. [35] Let be a BK-space. Then the following statements hold.

1. 1. then and T_L is compact if and only if
2. 2. then and T_L is compact if and only if
3. 3. If X has AK or and then − L_s − , where l = (l_t) and for each

Lemma 5.12. [35] Let be a BK-space. If then

and T_L is compact if and only if where is the sub-collection of consisting of all nonempty and finite subsets of with elements that are greater than n.

Theorem 5.12.

1. 1. If then holds.
2. 2. If then holds.
3. 3. If then holds.
4. 4. If then holds, where

Proof:

1. Let Since the series converges for each we have From Lemma 5.8, we have for each By using Lemma 5.11 (i), we deduce that
2. Let By Lemma 5.9, we have Hence, from Lemma 5.11 (iii), we have , where a = (a_t) and for each Moreover, Lemma 5.7 implies that for each
3. Let Since, for each from Lemma 5.11(ii), we have .
4. Let By Lemma 5.9, we have From Lemma 5.12, it follows that
   Moreover, Lemma 5.5 implies that .

□

Corollary 5.3.

1. 1. T_L is compact for if
2. 2. T_L is compact for if and only if
3. 3. T_L is compact for if and only if
4. 4. T_L is compact for if and only if where

Conclusion

Due to the vast application of quantum calculus, Motzkin numbers and difference operator in various mathematical and scientific disciplines, we have constructed generalized q-difference Motzkin sequence spaces , , and and explore their topological properties. We determine Schauder bases for and and compute α-, β- and γ-duals of the newly defined spaces. Further, we characterize some matrix mappings from the spaces and to the spaces . Lastly, compact operators are characterized on the spaces .

Acknowledgments

We thank Fujian Provincial Key Laboratory of Data-Intensive Computing, Fujian University Laboratory of Intelligent Computing and Information Processing and Fujian Provincial Big Data Research Institute of Intelligent Manufacturing of China.