Some problems on solid sequence spaces generated by infinite matrices

In this thesis, a study on some results, examples and problems related to the solid sequence spaces, denoted by 𝑠𝑙𝑑 − 𝐴 −1 (𝜆) , derived from infinite matrices, in particular, from some special kinds of infinite matrices like Cesaro and Toeplitz matrices and a solid sequence space 𝜆 is given. Let 𝐸 be a set of complex or real sequences and 𝐴 an infinite matrix with non-negative entries. By definition, 𝑥 ∈ 𝐴 −1 (𝐸) if and only if the sequence 𝑥 falls in the domain of 𝐴 and 𝐴𝑥 ∈ 𝐸. Similarly, we define , 𝑥 ∈ 𝑠𝑙𝑑 − 𝐴 −1 (𝐸) if and only if the sequence |𝑥| falls in the domain of 𝐴 and 𝐴|𝑥| ∈ 𝐸. Thus, if 𝐸 = 𝑙𝑝 (1 < 𝑝 < ∞ ) and 𝐴 is the Cesaro matrix, then 𝑠𝑙𝑑 − 𝐴 −1 (𝐸) becomes the classical Cesaro sequence space 𝑐𝑒𝑠𝑟𝑝. So solid sequence space, 𝑠𝑙𝑑 − 𝐴 −1 (𝐸) is the generalization of the classical Cesaro sequence space 𝑐𝑒𝑠𝑝. In this context, some results examples and open problems on this pullback sequence space and projective limits obtained by (Johnson and Mohapatra 1985), (Johnson and Polat 2015) ve (Johnson and Polat 2016) are considered. For example, some conditions under which the pullback spaces inherit properties of 𝐸 such as 𝐿𝐶𝐶 , 𝐴𝐾 and Hausdorffness properties or 𝐸 inherits properties of its pullbacks are established, and various counterexamples are given to show that many of the results are not satisfied. Also, it is investigated that whether projective limit is normable or not.