Generalized sequence spaces and matrix operators

Abstract

1. INTRODUCTION Studies on sequence spaces was further extended through Summability Theory. The summability theory originated from the attempts made by the mathematicians to give limits to the divergent sequences, on taking its transformation. O. Toeplitz was the first person to study the summability methods as a class of transformations of complex sequences by complex infinite matrices. It was followed by the works due to I. Schur, S. Mazur, W. Orlicz, K. Knopp, G. M. Petersen, S. Banach, G. Kothe and O. Toeplitz, are a few to be named. The works on paranormed sequence spaces was initiated by H. Nakano and S. Simons. It was further studied by I. J. Maddox, C. G. Lascarides, S. Nanda, D. Rath, G. Das, Z.U.Ahmed, B. Kuttner and many others. The scope for the studies on sequence spaces was extended due to the application of different techniques and notions of functional analysis. Throughout w, c, c0, l, c, c0, lp, y, y0, S0 will represent the classes of all, convergent, null, bounded, statistically, convergent, statistically null, p-absolutely summable, summable, summable to zero, bounded statistically null sequences with non-zero terms respectively. 2. OBJECTIVE OF THE STUDY The aim of the work carried is to introduce some new sequence spaces and study their different properties like completeness, solidity, separability, symmetricity etc. Further inclusion relations among the introduced sequences spaces and the introduced ones with some of the existing ones have been established. Some new matrix classes have been characterized. The first chapter of the thesis is introductory in nature. Most of the definitions and results have been collected in this chapter, those are used in the subsequent chapters of the thesis. Further the preliminaries of the works carried are given to have a clear picture of the background and the development of the topics on which the works have been carried in the thesis. 3. THE SUMMARY OF THE WORK DONE In order to extend the notion of convergence of sequences, statistical convergence of...