On the Domain of the Fibonacci Difference Matrix

Matrix Fˆ derived from the Fibonacci sequence was first introduced by Kara (2013) and the spaces lp(F) and l∞(F); (1 ≤ p < ∞) were examined. Then, Başarır et al. (2015) defined the spaces c0(F) and c(F) and Candan (2015) examined the spaces c(F(r,s)) and c0(F(r,s)). Later, Yaşar and Kayaduman (2018) defined and studied the spaces cs(F(s,r)) and bs(F(s,r)). In this study, we built the spaces cs(F) and bs(F). They are the domain of the matrix F on cs and bs, where F is a triangular matrix defined by Fibonacci Numbers. Some topological and algebraic properties, isomorphism, inclusion relations and norms, which are defined over them are examined. It is proven that cs(F) and bs(F) are Banach spaces. It is determined that they have the γ, β, α -duals. In addition, the Schauder base of the space cs(F) are calculated. Finally, a number of matrix transformations of these spaces are found.


Introduction
Cooke [1] formulated the theory of infinite matrices in the book "Infinite Matrices and Sequence Spaces".Many researchers have investigated infinite matrices after the publication of this book in 1950.In most of these studies, the domain of infinite matrices on a sequence space was studied.In this study, we address the question: What are the properties of the domain of the Fibonacci band matrix on sequence spaces bs and cs?The domain of the Fibonacci band matrix creates a new sequence space.We handle algebraic properties of this new space in order to determine its duals and its place among other known spaces, and to characterize the matrix transformations of this space.
One difficulty of this study is to determine whether the new space is the contraction or the expansion, or the overlap of the original space.Another difficulty is to determine the matrix transformations on this space and into this space.For the first problem, we give a few inclusion theorems.For the second problem, we use the matrix transformation between the standard sequence spaces and two theorems.
Generating a new sequence space and researching on its properties have been important in the studies on the sequence space.Some researchers examined the algebraic properties of the sequence space while others investigated its place among other known spaces and its duals, and characterized the matrix transformations on this space.
We can create a new sequence space by using the domain of infinite matrices.Ng-Lee [2] first investigated the domain of an infinite matrix in 1978.In the same period, Wang [3] created a new sequence space by using another infinite matrix.Many researchers such as Malkovsky [4], Altay, and Başar [5] followed these studies.This topic was studied intensively after 2000.
Leonardo Fibonacci invented Fibonacci numbers.He introduced Fibonacci numbers originated from a rabbit problem.These numbers create a number sequence: 1,1,2,3,5,8,13,21,34,55,89,....This sequence has important properties and applications in various fields.Let us indicate the Fibonacci sequence by (f n ).f n is defined as Let us indicate the set of all real-valued sequences with w and list some subspaces of w called standard sequence spaces.
Now let us take real valued infinite matrix T = (t nk ), where t nk is a real number for every n,k∈N Let A and B be sequence spaces.Sequence Tx = {T n (x)} is T-transform of a for every a = (a k )∈A.Here, Ta∈B and and T n (a)→t (t exists for every n∈N).Then, T is called a matrix transformation from A to B. Now let us take infinite matrix T and sequence space δ to define domain of infinite matrix T. The domain of the matrix T on δ is characterized by Many reserachers have studied the domain of a matrix on a sequence space.For more detailed information on these new sequence spaces, see references .
Let δ be a sequence space.The γ, β, α -duals of δ are defined, respectively, as follows In this study, spaces cs(F) and bs(F) are introduced and the related notations are given in Section 2. In addition, some topological and algebric properties, isomorphism, inclusion relations and norms which are defined over them are examined.The γ, β, α -duals of these spaces are determined in Section 3. The Schauder base of space cs(F) are calculated.Finally, many matrix transformations of these spaces are found.In the last section, the results and previous studies and the working hypotheses are discussed.
A detailed literature review was performed before this study was started.Scans were made on related articles, magazines, and books.As a result of these scans, the part related to our subject was synthesized and the results were noted.These results were then applied to our problem area.Finally, the results of this study were obtained.

The Domain of Fibonacci Difference Matrix F on Bounded and Convergent Series
In this section, cs(F) and bs(F) are introduced.Related notations are given.In addition, some topological and algebric properties, isomorphism, inclusion relations, and norms defined over them are examined.
Let spaces cs(F) and bs(F) be the domain of the matrix F on cs and bs, where F = {f nk } infinite matrix is defined by (f n ) for all k,n∈N.Then we inroduce cs(F) and bs(F) as We can see cs(F) = (cs) F and bs(F) = (bs) F by using Equation (2).
Let the inverse matrix of F be F −1 .For all k,n∈N, F −1 = {F −1 nk } is found as Let us take sequence x = (x n ).If y = Fx, then we calculate as Herefrom, if we calculate inverse of F, then we find that x = F −1 y and Now, let us give some theorems related to our study.
Theorem 1. bs(F) is a linear space.
Proof.The proof is left to the reader since it is easy to show.
Theorem 2. cs(F) is a linear space.
Proof.The proof is left to the reader since it is easy to show.
Theorem 3. bs(F) is a normed space with: Proof.The proof is left to the reader since it is easy to show.
Proof.The proof is left to the reader since it is easy to show.
Proof.Let us take T: bs(F) →bs mentioned Equation ( 4) by x→y = Tx = Fx.It is easy to see that T is linear and injective.
We must find T is surjective.Let y = (y n )∈bs.By using Equation ( 5) and Equation ( 6), we see We see that x∈bs(F).Hence, T is surjective.In addition, bs(F) and bs izometric because x bs(F) = y bs .Theorem 6. cs(F) is isomorphic to cs.
Proof.The proof can be made similar to Theorem 5, so it is left to the reader.Theorem 7. bs(F) is a Banach space with Equtaion (6).
Proof.It is easy to see the norm conditions are ensured.Let a Cauchy sequence x i = (x k i ) in bs(F) for each i∈N.For all k∈N, we have from Equation ( 4).For all ε > 0 there is n 0 = n 0 (ε) such that for all i,m ≥ n 0 .y i →y (i→∞) such that y ∈ bs exists, since bs is complete.Since bs and bs(F) are isomorphic, bs(F) is complete.It hereby is a Banach space.
Proof.It is easy to see the norm conditions are ensured.Let a Cauchy sequence x i = (x k i ) in cs(F) for each i∈N.For all k∈N, we have from Equation ( 4).For all ε > 0, there is n 0 = n 0 (ε) such that for all i,m ≥ n 0 .y i →y (i→∞) such that y ∈ cs exists, since cs is complete.Since cs and cs(F) are isomorphic, cs(F) is complete.It hereby is a Banach space.Now, let R = (r nk ) infinite matrix.Let us list the following: sup sup sup sup sup sup The collection of all finite subsets of N denoted by F .
Proof.Suppose x∈bs.If we show that F is an element of (bs,bs) then x is element of bs(F).For this, F must provide Equations ( 8) and (9).Since lim k f nk = 0 for each n ∈ N, Equation ( 8) is provided.
Proof.Suppose x∈cs.If we show that F is element of (cs,cs) then x is element of cs(F).For this, F must provide Equations ( 10) and ( 9).Equation ( 9) has been provided from the Theorem 10.If we look at the Equation ( 10) then, for each k∈N, such that l∈C exists.
is a base for cs(F).Every x ∈ cs(F) can write as a single x = ∞ ∑ k=0 y k u k such that y k = ( Fx) k .

The Duals of cs(F) and bs(F) and Matrix Transformations
Let us give the two lemmas to use in the next stage.
Lemma 16.Let infinite matrix C = (c nk ) 1 2 and a = (a n ) ∈ w.Let us take C = aF −1 , that is, Proof.Let x = (x n ) and a = (a n ) elements of w. y = (y n ) such that y = Fx which is defined in Equation ( 4).

Lemma 17. [41]
Let us take a = (a k )∈w and infinite matrix C = (c nk ).Let the inverse matrix H = (h nk ) of the triangular matrix G = (g nk ) is given by Then, for any sequence space δ, If we consider Lemma 9, Lemma 16 and Lemma 17 together, the following is obtained; Corallary 18.Let us take r = (r k )∈ w and infinite matrix A = (a nk ) and B = (b nk ) such that If we take d 1 , d 2 , d 3 , d 4 , d 5 , d 6 , d 7 and d 8 as follows: where and for all k, m, n ∈ N.
Let us give almost convergent sequences space, which was first defined by Lorentz [42].Let It is denoted by ĉ − limt = α.In addition, ĉs and ĉ0 mean the spaces of almost convergent series and almost null sequences, respectively.ĉ0 and ĉ are Now, let us take infinite matrix R = (r nk ) and list the following: sup sup Lemma 21.Let infinite matrix R = (r nk ) for all k,n∈N.Then, (1) R = (r nk ) ∈ ( ĉ, cs) iff Equation (23) and Equations ( 37)-( 39) hold [43].

Discussion
Kızmaz [47] first introduced the difference sequence operator in 1981.Generalized difference sequence spaces were characterized and investigated by Kirişçi and Başar [4] in 2010.Kara [27] first defined the Fibonacci Difference Matrix F, which created the Fibonacci sequence (f n ) in 2013.He also introduced the new sequence spaces p (F) and ∞ (F); where 1 ≤ p < ∞.The spaces c(F(r,s)) and c 0 (F(r,s)) were introduced by Candan [28] in 2015.In 2015, the sequence space p (F(r,s)) was introduced and studied by Candan and Kara [19]; where 1 ≤ p ≤ ∞.In addtion, a class of compact operators on p (F) and ∞ (F) was characterized by Kara et al. [32], where 1 ≤ p < ∞.
In the present study, we introduced the domain of a triangular infinite matrix on a sequences space.We described spaces cs(F) and bs(F), where F, cs, and bs are the Fibonacci Difference Matrix, convergent and bounded series, respectively.It was demonstrated that bs(F) are the linear spaces, and given that cs(F) is linear space in Theorem 6. without proof and, they have the same norm where x∈cs(F) or x∈bs(F).It was found that they are Banach spaces.In addition, inclusions theorems were examined and found.Finally, the γ, β, α -duals of them were calculated.Finally, some matrix transformations as a main result were given.

Theorem 13 .Corallary 15 .
cs(F) ⊃ c is not valid.Proof.Let x = (x k ) = (f 2 k+1 ).Then y = Fx = (1,0,0....)∈cs.On the other hand, f 2 k+1 →∞ as k→∞.It is clear x∈cs(F), but x / ∈ c.This result completes the proof.Theorem 14. cs(F) ⊂ bs(F) is valid.Proof.If x∈cs(F), y = Fx∈cs.Hence, ∑ k Fx ∈ c.Since c⊂ ∞ , ∑ k Fx ∈ l ∞ .Hence, Fx∈bs.That is, x∈bs(F).This result completes the proof.Let us take normed space A and let (a k ) ∈ A. If there is only one scalar sequence (v k ) such that k = 0 then (a k ) is called a Schauder base for A. Now, let us give corallary releated to Schauder basis.Let a sequence u (k) = u (k) n n∈N in cs(F) be for each k ∈ N and u