ON THE SPECTRA OF N ¨O RLUND Q AND ALMOST N¨O RLUND Q OPERATORS

Abstract

The main objectives in this study are to investigate and determine the spectra ofN¨orlund means acting as operators on Banach sequence spaces. We also aim todetermine the spectra of almost triangular matrices. Specifically we determine thespectrum of the N¨orlund Q operator on the Banach spaces c0, c, bv0, bv. We alsodetermine the spectrum of an almost N¨orlund Q matrix operator on c0 and c.In all the cases mentioned we show that the spectrum comprises of the disccentered at the point ( 12 ,0) of radius 12 . We also construct the fine spectrum of the Qoperator on c.Apart from the more obvious benefit i.e., the solution of systems of linear equationof which the spectrum of operators is all about; there are other more subtle, butequally important applications of the research. A central problem in the whole ofmathematics and even science and engineering; is the determination of the convergenceor non- convergence of sequences and series. Mathematics, especially Mathematicalanalysis, develops and is maintained via the concept of convergence of sequencesand series. Even in applied science and Engineering, one is interested in theconvergence of a sequence or a series of results generated during experimentation.Established theorems such as the ratio theorems and integral theorem, are not applicablein a variety of sequences and series. Even where they apply, they just determineconvergence and not the limit or sum of the convergent sequence or series. Tauberiantheorems in Summability Theory handle this problem well.The convergence andeven limit of a convergent sequence or series is determined from the convergence ofsome transform of it together with a side condition, (Maddox, 1970); (Boos, 2000)pp. 167 - 204; (Hardy, 1948) pp. 148 - 177; (Powell and Shah, 1972) pp. 75 - 92; or(Maddox, 1980) pp. 65 - 80, e.t.c. The spectrum of an operator plays a crucial rolein the development of a Tauberian theory for the operator, (Dunford and Schwartz,1957) pp. 593 - 597. It is evident from the mentioned books that a Tauberian theoryfor N¨orlund operators is almost non-existent. Therefore we are confident the resultsxdeveloped in the thesis will open a floodgate for such theorems for N¨orlund means. Inturn this will find application in diverse fields such as, integral transforms and Fourieranalysis; and in probability and statistics through such areas involving central limittheorem, 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 toour study.Chapter II deals with the spectrum of the Q matrix on c0 and c. In chapter IIIwe 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. InChapter V we investigate the spectrum of an almost N¨orlund Q matrix operator onc0 and c. Chapter VI gives an overview of the results obtained and points the wayforward for future research interests.In achieving the results, we used a combination of classical and modern functionalanalytic methods as well as Summability methods. Functional analytic methods usuallyappeal to the powerful Banach space theorems, such as Hahn - Banach; Banach-Steinhaus; exetra. Classical Summability methods employ sequence space mappingtheorems such as Silverman - Toeplitz; Kojima - Shur; exetra