On the Spectra OF N ¨O RLUND Q and almost N¨O RLUND Q operators

Abstract

operator, (Dunford and Schwartz, 1957) pp. 593 - 597. It is evident from the mentioned books that a Tauberian theory for N¨orlund operators is almost non-existent. Therefore we are confident the results x developed in the thesis will open a floodgate for such theorems for N¨orlund means. In turn this will find application in diverse fields such as, integral transforms and Fourier analysis; and in probability and statistics through such areas involving central limit theorem, almost sure convergence, summation of random series, Markov chains e.t.c; (Boos, 2000) pp. 256 - 257. Chapter 1 deals with literature review, a summary of Functional Analysis material; as well as classical summability methods; especially those that are pertinent to our study. Chapter II deals with the spectrum of the Q matrix on c0 and c. In chapter III we investigate the spectrum of the N¨orlund Q operator on the spaces bv0 and bv. Chapter IV is concerned with the fine spectrum of the Q matrix operator on c. In Chapter V we investigate the spectrum of an almost N¨orlund Q matrix operator on c0 and c. Chapter VI gives an overview of the results obtained and points the way forward for future research interests. In achieving the results, we used a combination of classical and modern functional analytic methods as well as Summability methods. Functional analytic methods usually appeal to the powerful Banach space theorems, such as Hahn - Banach; Banach- Steinhaus; exetra. Classical Summability methods employ sequence space mapping theorems such as Silverman - Toeplitz; Kojima - Shur; exetra.