On Some New Generalized Difference Sequence Spaces of Nonabsolute Type

We define a new triangle matrix by the composition of the matrices and . Also, we introduce the sequence spaces , and by using matrix domain of the matrix on the classical sequence spaces , and , respectively, where . Moreover, we show that the space is norm isomorphic to for . Furthermore, we establish some inclusion relations concerning those spaces and determine -, -, and -duals of those spaces and construct the Schauder bases , and . Finally, we characterize the classes of infinite matrices where and . 1. Preliminaries, Background, and Notation By a sequence space, we understand a linear subspace of the space of all complex sequences which contains , the set of all finitely nonzero sequences. We write , and for the classical sequence spaces of all bounded, convergent, null, and absolutely -summable sequences, respectively, where . Also by and , we denote the spaces of all bounded and convergent series, respectively. We assume throughout unless stated otherwise that with and use the convention that any term with negative subscript is equal to zero. We denote throughout that the collection of all finite subsets of by . Let be an infinite matrix and two sequence spaces. Then, defines a matrix mapping from to and is denoted by if for every sequence the sequence , the -transform of , is in , where By , denote the class of all matrices such that . Thus, if and only if the series on the right hand side of (1) converges for each and , and we have for all . A sequence is said to be -summable to if converges to , which is called the -limit of . A matrix is called a triangle if for and for all . It is trivial that holds for the triangle matrices and a sequence . Further, a triangle matrix has a unique inverse which is also a triangle matrix. Then, holds for all . Let us give the definition of some triangle limitation matrices which are needed in the text. Let be a sequence of positive reals and write Then the Cesàro mean of order one, Riesz mean with respect to the sequence , and Euler mean of order with are, respectively, defined by the matrices ,？？ , and , where for all . We write for the set of all sequences such that for all . For , let . Let , and define the summation matrix , the difference matrix , and the generalized weighted mean or factorable matrix ,？？ ,？？ by for all , where and depend only on and , respectively. Let and be nonzero real numbers, and define the generalized difference matrix by for all . We note that if we choose and then the matrix is reduced to the backward difference. For a sequence space , the matrix domain of an infinite matrix