Studies on some sequence spaces and characterization of some matrix classes

Abstract

1. INTRODUCTION Studies on sequence spaces were further extended through Summability Theory. The 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 then followed by the works due to I. Schur, S. Mazur, W. Orlicz, K. Knopp, G. M. Petersen, S. Banach and G. Kothe 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. Introducing the notion of statistical convergence of sequences by H. Fast, R. C. Buck and I. J. Schoenberg independently extended the notion of convergence of sequences. It was further investigated from sequence space point of view and linked with summability by J. A. Fridy, C. Orthan, T. Salat and many others. Throughout w, c, c0, l, lp, l1, c, co, c1 and c10 will represent the classes of all, convergent, null, bounded, p-absolutely summable, absolutely summable, statistically convergent, statistically null, I-convergent, I-null sequences respectively. We write m=cnl; m1=c1nl; m0=c0nl and m10=c10nl. Further So, So, S1o will represent the subsets of the spaces co, mo and m1o respectively with non-zero terms. Further w(X), c(X), c0(X), l(X), lp(X), c(X), c0(X) will denote the spaces of all, convergent, null, bounded, p-absolutely summable, statistically convergent and statistically null X-valued sequence spaces 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, monotonicity etc. Further inclusion relations...