Sequence spaces derived by the triple band generalized Fibonacci difference operator

In this article we introduce the generalized Fibonacci difference operator F(B) by the composition of a Fibonacci band matrix F and a triple band matrix B(x, y, z) and study the spaces k(F(B)) and ∞(F(B)). We exhibit certain topological properties, construct a Schauder basis and determine the Köthe–Toeplitz duals of the new spaces. Furthermore, we characterize certain classes of matrix mappings from the spaces k(F(B)) and ∞(F(B)) to space Y ∈ { ∞, c0, c, 1, cs0, cs,bs} and obtain the necessary and sufficient condition for a matrix operator to be compact from the spaces k(F(B)) and ∞(F(B)) to Y ∈ { ∞, c, c0, 1, cs0, cs,bs} using the Hausdorff measure of non-compactness.


Introduction
Throughout this paper, the set of all real valued sequences shall be denoted by w. Any linear subspace of w is known as a sequence space. The sets k (k-absolutely summable sequences), ∞ (bounded sequences), c 0 (null sequences) and c (convergent sequences) are a few examples of classical sequence spaces. Moreover, cs and bs will represent the spaces of all convergent and bounded series, respectively. Here and in what follows 1 ≤ k < ∞, unless stated otherwise. A Banach space having continuous coordinates is known as BK -space. The spaces k and X = { ∞ , c, c 0 } are BK -spaces endowed with the norms s k = ( ∞ v=0 |s v | k ) 1/k and s ∞ = sup v∈N |s v |, respectively. The theory of matrix mappings plays an important role in summability theory because of its well-known property 'a matrix mapping between BK -spaces is continuous [6,47]' . Let X and Y be any two sequence spaces and = (ψ rv ) be an infinite matrix of real entries. The notation r shall mean the sequence in the rth row of the matrix . Furthermore, the sequence s = {( s) r } = { ∞ v=0 ψ rv } is called the -transform of the sequence s = (s r ) ∈ X, provided that the series ∞ v=0 ψ rv exists. Furthermore, if, for each sequence s in X, itstransform is in Y, then we say that is a matrix mapping from X to Y. We shall denote the family of all matrices that map from X to Y by (X : Y).
Define the set X = {s ∈ w : s ∈ X}. (1.1) The set X is a sequence space and is known as the domain of matrix in the space X.

Fibonacci sequence spaces
Fibonacci numbers are also considered to be Nature's numbers. They can be found everywhere around us, from the leaf arrangements in plants, to the pattern of the florets of flowers, the bracts of pinecones or the scales of pineapple. The number sequence 1, 1, 2, 3, 5, 8, . . . is called the Fibonacci sequence. Note that any number in the sequence is the sum of the two numbers preceding it. Thus, if {f v } ∞ v=0 is the sequence of Fibonacci numbers, then The ratio of the successive terms in the Fibonacci sequence approaches an irrational number 1+ √ 5 2 , which is called the golden ratio. This number has great application in the field of architecture, science and arts. Some more basic properties of Fibonacci numbers [37] can be listed as follows: The Fibonacci double band matrix F = (f rv ) is defined by [29] Kara [29] introduced the sequence spaces k (F) = ( k ) F and ∞ (F) = ( ∞ ) F . Later on, Başarır et al. [7] studied Fibonacci difference spaces c 0 (F) = (c 0 ) F and c(F) = (c) F . Since then many authors studied and generalized Fibonacci difference sequence spaces. We refer to [11,13,14,16,[30][31][32][33][34] for relevant studies. Motivated by the above studies, we introduced generalized Fibonacci difference operator by the composition of the Fibonacci band matrix F and the triple band matrix B(x, y, z). We study the domains k (F(B(x, y, z))) and ∞ (F(B(x, y, z))) of the matrix operator F(B(x, y, z)) in the spaces k and ∞ , respectively, investigate certain topological properties of the spaces and construct the Schauder basis of the sequence space k (F(B(x, y, z))). In Sect. 4, we obtain the Köthe-Toeplitz duals of the sequence spaces k (F(B(x, y, z))). In Sect. 5, we characterize certain classes of matrix mappings from the spaces k (F(B(x, y, z))) and ∞ (F(B(x, y, z))) to the space Y, where Y ∈ { ∞ , c, c 0 , 1 , cs 0 , cs, bs}. In Sect. 6, we characterize certain classes of compact operators on the spaces k (F(B(x, y, z))) and ∞ (F(B(x, y, z))) using the Hausdorfff measure of non-compactness (or in short Hmnc).

Main results
In the present section, we introduce the product matrix F(B), where B = B(x, y, z) is the triple band difference matrix, and obtain its inverse and introduce generalized Fibonacci difference sequence spaces k (F(B)) and ∞ (F(B)), exhibit certain topological properties of these spaces and obtain basis of the space k (F(B)).
Combining Fibonacci band matrix F and difference operator B, the product matrix F(B) = (f (B)) rv is defined by  Equivalently, F(B) can also be expressed as Now we obtain the inverse of the product matrix F(B).

Lemma 3.2 ([7])
The Fibonacci band matrix F has the inverse F -1 defined by

Lemma 3.3 The inverse of the product matrix F(B) is defined by the triangle
Proof The result follows from Lemma 3.1 and Lemma 3.2.
Define the sequence t = (t v ) in terms of the sequence s = (s v ) by Note that the terms with negative subscripts is considered to be zero. The sequence t is called F(B)-transform of the sequence s. Now we define the spaces k (F(B)) and ∞ (F(B)) by The spaces k (F(B)) and ∞ (F(B)) may be redefined in the notation of (3.2) as We further emphasize that the spaces k (F(B)) and ∞ (F(B)) are reduced to certain classes of sequence spaces in the special cases of x, y, z ∈ R.

Theorem 3.4
The spaces k (F(B)) and ∞ (F(B)) are BK -spaces endowed with the norms defined by respectively.
Proof The proof is a routine exercise and hence is omitted.
Proof We present the proof for the space k (F(B)). It is clear that the mapping T : s is linear and one-one. Let t = (t r ) ∈ k define the sequence s = (s r ) by Then we have This implies that s ∈ k (F(B)). Thus we realize that T is onto and norm preserving. Thus To end this section, we construct a sequence that forms a Schauder basis for the space k (F(B)). We recall that a Schauder basis in a normed space X is a sequence s = (s r ) such that to every element u in X there corresponds a unique sequence of scalars (a r ) satisfying Let e (v) denote the sequence with 1 in the vth position and 0 elsewhere. We are well aware that the set {e (v) : v ∈ N} is a Schauder basis of the space k . Moreover, the mapping T defined in Theorem 3.5 is onto, therefore the inverse image of the set {e (v) } forms the basis of the space k (F(B)). This statement gives us the following result.
for each r ∈ N. Then the sequence (c (v) ) is a Schauder basis for the space k (F(B)) and every s ∈ k (F(B)) can be uniquely expressed in the form Proof The result follows from Theorems 3.4 and 3.6.

Köthe-Toeplitz duals (or α-, βand γ -duals)
In present section, we determine Köthe-Toeplitz duals of the space k (F(B)) and ∞ (F(B)). It is to mention that we have not provided the proof for the case k = 1 as the proof is similar to the case 1 < k ≤ ∞. The proofs are provided only for the latter case.
where the matrix D = (d rv ) is defined by (1) .
Proof The proof is analogous to the proof of previous theorem except that Lemma 4.4 is employed instead of Lemma 4.2.

Matrix mappings
In the present section, we characterize certain class of matrix mappings from the spaces k (F(B)) and ∞ (F(B)) to the space Y ∈ { ∞ , c, c 0 , 1 , bs, cs, cs 0 }. The following theorem is fundamental in our investigation. Theorem 5.1 Let 1 ≤ k ≤ ∞ and X be any arbitrary subset of w. Then = (ψ rv ) ∈ ( k (F(B)) : X) if and only if (r) = (φ (r) mv ) ∈ ( k : c) for each r ∈ N, and = (ψ rv ) ∈ ( k : X), where Proof The result immediately follows from the proof of Theorem 4.1 of [35]. Hence we omit the details. Now, using the results presented in Stielglitz and Tietz [60] together with Theorem 5.1, we obtain the following results. also holds.

Hausdorff measure of non-compactness (Hmnc)
In the current section, B(X) shall denote the unit ball in X. The notation B(X : Y) represents the family of all bounded linear operators acting from Banach spaces X to Y, which itself is a Banach space endowed with the operator norm C = sup s∈B(X) Cs . We denote for ς ∈ w, provided that the series on the right hand side of (6.1) exists. One may clearly observe that ς ∈ X β . Furthermore, the operator C is said to be compact if the domain of X is all of X and for every bounded sequence (s r ) in X, the sequence ((Cs) r ) has a convergent subsequence in Y.
The Hmnc of a bounded set J in a metric space X is defined by B(s l , n l ), s l ∈ X, n l < ε (l = 0, 1, 2, . . . , r), r ∈ N , where B(s l , n l ) represents unit ball with centre s l and radius n l and l = 0, 1, 2, . . . , r.
Hmnc is an important tool that determines the compactness of an operator between BKspaces. An operator C : X → Y is compact if and only if C χ = 0, where C χ represents Hmnc of the operator C and is defined by C χ = χ(C(B(X))). Using Hmnc, several authors obtained necessary and sufficient conditions for matrix operators to be compact between well-known BK -spaces. For relevant literature, one may refer to [2,13,40,[49][50][51][52]. The reader may also consult the recent publications [22,24,25,53,62], which are related to compact operators and Hmnc in BK -spaces.
Before proceeding to the main results of this section, we list certain well-known results that are crucial in finding our result below.
where I X is the identity operator on X.  for r ∈ N. Thus by applying Part (a) of Lemma 6.5, we immediately get the desired result. (F(B)) : c), then using Lemma 6.1, we have ∈ ( k : c). Then applying Part (b) of Lemma 6.5, we get Thus, we realize that The proof is analogous to the proof of Part (a) of Theorem 6.9 except that we employ Part (c) of Lemma 6.5 instead of Part (a) of Lemma 6.5. (g) The proof is analogous to the proof of Part (e) of Theorem 6.9 except that we employ Part (c) of Lemma 6.7 instead of Part (a) of Lemma 6.7. Corollary 6.10 Let 1 < k < ∞. Then the following results hold: where φ = (φ v ) and φ v = lim r→∞ φ rv for each v ∈ N. (f ) If ∈ ( ∞ (F(B)) : cs), then (F(B)) : bs), then 0 ≤ C χ ≤ lim sup r→∞ ( ∞ v=0 | r m=0 φ mv |).
Proof The proof is analogous to the proof of Theorem 6.9.