SOME MATRIX AND COMPACT OPERATORS OF THE ABSOLUTE FIBONACCI SERIES SPACES

converges for all n ∈ N = {0, 1, 2, . . .}, then, by A(x) = (An(x)), we denote the Atransform of the sequence x = (xv). Also, we say that A defines a matrix transformation from X into Y , and denote it by A ∈ (X, Y ) or A : X → Y if Ax = (An(x)) ∈ Y for every x ∈ X. The α-, β-, γduals of X and the domain of the matrix A in X are defined by X = { ∈ ω : ( nxn) ∈ ` for all x ∈ X} , X = { ∈ ω : ( nxn) ∈ cs for all x ∈ X} ,


Introduction
Let ω be the set of all sequences of complex numbers. We write c, ∞ , c s , b s and k , k ≥ 1, for the sequence space of all convergent, bounded sequences; for the spaces of all convergent, bounded, k-absolutely convergent series, respectively. Let X and Y be two subspaces of ω and A = (a nv ) be an arbitrary infinite matrix of complex numbers. If the series X γ = { ∈ ω : ( n x n ) ∈ b s for all x ∈ X} and (1.1) respectively. Further, X is said to be a BK-space if it is a complete normed space with continuous coordinates p n : X → C defined by p n (x) = x n for all n ∈ N. If there exists unique sequence of coefficients (x k ) such that, for each x ∈ X, then, the sequence (b k ) is called the Schauder basis (or briefly basis) for a normed sequence space X, and in this case we write x = ∞ k=0 x k b k . For instance, the sequence (e (j) ) is the Schauder basis of the space k , where e (j) is the sequence whose only nonzero term is 1 in jth place for each j ∈ N.
Now take x v as an infinite series with nth partial sum s n and let (u n ) be a sequence of positive terms. Then, the series x v is said to be summable |A, u n | k , k ≥ 1, if (see [32]) where ∆A n (s) = A n (s) − A n−1 (s), A −1 (s) = 0. Note that this method includes some well known methods. For example, if A is the matrix of weighted mean N , p n (resp. u n = P n /p n ), then it reduces to the summability N , p n , u n k [36] (the summability N , p n k [10]). Also if we take A as the matrix of Cesàro mean of order α > −1 and u n = n, then we get summability |C, α| k in Flett's notation [11].
A large literature has recently grown up, concerned with producing sequence spaces by means of matrix domain of a special limitation method and studying their algebraic, topological structure and matrix transformations (see [1-7, 15-18, 25]). Also, some series spaces have been derived and studied by absolute summability methods from a different point of view (see [9-14, 23-26, 28-34, 36]). The aim of this paper is to define the space |F u | k combining absolute summability and Fibonacci matrix given by Kara [15], investigate some inclusion relation, construct their α-, β-, γ-duals, basis and characterize some matrix operators related to that space, and also determine their norms and Hausdroff measures of noncompactness.
Firstly, we mention some properties of Fibonacci numbers as follows: the sequence (f n ) of Fibonacci numbers is given by the relations that is, each term is equal to the sum of the previous two terms. The sequences of Fibonacci numbers have been important for artist, architects, physicists and mathematicians since the old. The ratio of Fibonacci numbers converges to the golden ratio which is one of the most interesting irrationals having an important role in number theory, algorithms, network theory, etc. Also, Fibonacci numbers have the following properties [19]: Fibonacci matrix F = (f nv ) has recently been defined by Kara [15] as follows: where f n be the nth Fibonacci number for every n ∈ N. Note that if we take the Fibonacci matrix instead of A, then |A, u n | k summability reduces to the absolute Fibonacci summability. On the other hand, since (s n ) is a sequence of partial sum of the series x v , we get (f nn +f n,n−1 )x j and so, 0, j > n. Now, we introduce the absolute Fibonacci space as follows: Also, it may be written that and k * is the conjugate of k, i.e., 1/k + 1/k * = 1 for k > 1, and 1/k * = 0 for k = 1.
With these matrices T = (t nv ) and E (k) = (e (k) nv ), according to the notation (1.1), it is obvious that |F u | k = ( k ) E (k) •T . Further, since every triangle matrix has a unique inverse which also is a triangle [37], T and E (k) have a unique inverseT = (t nv ) and (1.4) respectively.
Before the main theorems, we point out some well known lemmas which are needed in the proofs of theorems.
where F denotes the collection of all finite subsets of N.
1 exposes a rather difficult condition to apply in applications. So the following lemma is more useful in many cases, which gives equivalent norm.

The Hausdorff Measure of Noncompactness
If S and H are subsets of a metric space (X, d) and, for every h ∈ H, there exists domain is all of X and, for every bounded sequence (x n ) in X, the sequence (L(x n )) has a convergent subsequence in Y . We denote the class of such operators by C(X, Y ). If Q is a bounded subset of the metric space X, then the Hausdorff measure of noncompactness of Q is defined by and χ is called the Hausdorff measure of noncompactness.
The following lemma is very important to calculate the Hausdorff measure of noncompactness of a bounded subset of the space k .

Lemma 2.1 ([27]). Let Q be a bounded subset of the normed space
Let X and Y be Banach space and χ 1 and χ 2 be Hausdorff measures on X and Y , the linear operator L : X → Y is said to be (χ 1 , χ 2 )-bounded if L(Q) is a bounded subset of Y and there exists a positive constant M such that χ 2 (L(Q)) ≤ M χ 1 (L(Q)) for every bounded subset Q of X. If an operator L is (χ 1 , χ 2 )-bounded, then the number is called the (χ 1 , χ 2 )-measure noncompactness of L. In particular, if χ 1 = χ 2 = χ then we write L (χ,χ) = L χ .

Lemma 2.2 ([22]). Let X and Y be Banach spaces and
and

Absolute Fibonacci Space |F u | k
In this section, we investigate some inclusion relations, topological and algebraic structures of the space |F u | k . Also we characterize some classes of compact matrix operators on that space and compute their norms and Hausdroff measure of noncompactness.
Firstly, since |F u | k is generated from k , to explain a relation between the spaces k and |F u | k , we begin with the following theorem. Proof. To prove the inclusion k ⊂ |F u | k , it is sufficient to show that The proof is clear for the case k = 1, and so it is omitted. Let k > 1. Then, since the series applying Hölder's inequality, we get which completes the proof.
Proof. Take x ∈ |F u | k . Since k ⊂ q , then u 1 k * n n j=0 σ nj x j ∈ q and also, since u n ≤ M for all n ∈ N, where k * and q * are the conjugate of exponent of k and q, respectively. So this gives that x ∈ |F u | q , which completes the proof.
Proof. We note that k is a BK-space for 1 ≤ k < ∞. Further, since E (k) • T is a triangle matrix, it follows from Theorem 4.3.2 of [37], |F u | k = ( k ) E (k) •T is a BKspace. Since the sequence (e (j) ) is the Schauder basis of the space k , it can be written from Theorem 2.3 in [14] that b (j) = ( T n ( E (k) (e (j) ))) is a Schauder basis of the space |F u | k .
Proof. To prove the theorem, we should show that there exists a linear bijection between the spaces |F u | k and k where 1 ≤ k < ∞. Let consider the transformations 3) and (1.4). Since the matrices corresponding these transformations are triangles, it can be easily seen that T and E (k) are linear bijections. So, the composite function E (k) • T is a linear bijective operator. Furthermore, i.e., it preserves the norm. So the proof is completed.
In the following theorems, for the simplicity of presentation we take ξ vr v exists for all r , Theorem 3.5. Let 1 < k < ∞ and u = (u n ) be a sequence of positive numbers. Then,

By (1.3) and (1.4), it can be seen immediately that
where H = (h mr ) is defined by Therefore, ∈ {|F u | k } β if and only if H ∈ ( k , c). Applying Lemma 1.4 to the matrix H, we get {|F u | k } β = D 1 ∩ D 2 , which completes the proof. The proofs of other parts can similarly be proved, so we omit.
Theorem 3.6. Let 1 ≤ k < ∞, A = (a nv ) be an infinite matrix of complex numbers for each n, v ∈ N and define the matrix Further, letB = (b nv ) be a matrix given byb nv = lim and if and only if (a nv ) ∞ v=0 ∈ {|F u |} β and A(x) ∈ |F u | k for all x ∈ |F u |. Now, it can be easily seen from Theorem 3.5, (a nv ) ∞ v=0 ∈ {|F u |} β if and only if (3.1) and (3.2) hold. On the other hand, if a matrix R = (r nv ) ∈ ( , c), then the series R n (x) = ∞ v=0 r nv x v converges uniformly in n, because, the remaining term of the series tends to zero uniformly in n, since So we obtain Besides, according to Theorem 3.4, since |F u | k ∼ = k for 1 ≤ k < ∞, it follows that . Also, it is clear that the terms of matrix B can be expressed asb Hence, applying Lemma 1.3 to the matrix B, we have (3.3), which completes the first part of the proof.
Also, if A ∈ (|F u | , |F u | k ), then, since the spaces |F u | k and |F u | are BK-spaces, it is a bounded operator. In order to determine the operator norm of A, consider the isomorphisms T : Finally, assume that Q is a unique ball in |F u |.
This completes the proof.
By Theorem 3.6 and Lemma 2.2, the compact operators in this class are characterized as follows. and

Proof.
A ∈ (|F u | k , |F u |) if and only if A n = (a nv ) ∞ v=0 ∈ {|F u | k } β and A(x) ∈ |F u | where x ∈ |F u | k . By Theorem 3.5, it can be easily seen that A n ∈ {|F u | k } β if and only if (3.5) and (3.6) hold. Also, if any matrix R = (r nv ) ∈ ( k , c), then the series R n (x) = ∞ v=0 r nv x v converges uniformly in n. Because, the remaining term of the series tends to zero uniformly in n, since and so, it can be written that Since |F u | k ∼ = k for 1 ≤ k < ∞, by the Theorem 3.4, then, A(x) ∈ |F u | for every x ∈ |F u | k if and only ifH(z) ∈ |F u |, i.e., H(z) = E (1) • T •H(z) ∈ for every z ∈ k , where z = E (k) • T (x). This means that H ∈ ( k , ). Thus applying Lemma 1.2 to the matrix H, we get (3.7). This completes the proof of first part.
Since |F u | k is BK-spaces for every k ≥ 1, A is a bounded operator by Theorem 4.2.8 of [37].
Additionally, as Theorem 3.4, it can be written that A = T • E (1) • H • E (k) • T and so, .
Finally, let Q = S |Fu| . Since E (1) • T • AQ = H • E (k) • T Q, it follows by Lemma 2.1, Lemma 2.3 and Lemma 1.2 that which completes the proof.
Also, the compact operators can immediately be characterized by Lemma 2.2 and Theorem 3.7 as follows.