Some Matrix and Compact Operators of the Absolute Fibonacci Series Spaces

Abstract

In the present paper, we introduce the absolute Fibonacci space |F u | k , give some inclusion relations and investigate topological and algebraic structure such as BK-space, α-, β-, γ-duals and Schauder basis. Further, we characterize certain matrix and compact operators on these spaces, also determine their norms and Hausdroff meausures of noncompactness.

Characterizations of Matrix and Compact Operators on BK Spaces

“…So, there has been a large literature, concerned with characterizing completely all matrices which transform one given sequence space into another. Besides this, the literature has been also grown up in terms of the studies of many sequence spaces defined as domain of special matrices and related matrix operators (see, for instance, [1][2][3][4][6][7][8][9][10][11][12]). For a recent paper [1], the infinite matrix classes E r µ , E r µ q and E r µ q , E r µ have been introduced.…”

“…To mention few of them are [2][3][4][5][6]. On the one hand, from a different perspective, using the notion of absolute summability, a lot of new spaces of series summable by the absolute summability methods have studied and introduced by authors (see [1,[7][8][9][10][11][12][13][14]). In recent paper [1], the infinite matrix classes E r µ , E r µ q and E r µ q , E r µ have been investigated.…”

“…For instance, for Cesàro matrix with µ n = n and the weighted mean matrix, it reduces to the absolute Cesàro summability due to Flett [7] and the absolute weighted summability given by Sulaiman [6], respectively. For more applications, we refer readers to ( [1,[8][9][10]12]). Also, if we choose the Euler matrix E r = (e r ni ) instead of U, the summability |U, µ n | q is reduced to the absolute Euler summability |E r , µ n | q of order r. Here the terms of the matrix E r = (e r ni ) is given by…”

Absolute Lucas Spaces with Matrix and Compact Operators

“…Also if we take A as the matrix of Cesàro mean of order α > −1 and φ n = n, then we get the summability |C, α| p in Flett's notation [5]. The choice of the Fibonacci matrix instead of A leads to the absolute Fibonacci summability method [7]. In addition to the aforementioned spaces, several absolute series spaces have also taken place in the literature (see [6,8,19,25,[27][28][29]).…”

“…Using the Hausdorff measure of noncompactness, some compact operators on various sequence spaces are characterized by many authors. For example, Mursaleen and Noman in [21,22], Malkowsky and Rakocevic in [17] have used the Hausdorff measure of noncompactness method to characterize the class of compact operators on some known spaces, (see also [7,8,15,26]).…”