On the Spectrum of A Class of N ¨ORLUND Operators

Abstract

Spectral theory is an important branch of Mathematics due to its application in other branches of science. In summability theory, different classes of matrices have been investigated and characterized. There are various types of summability methods e.g. N ¨ orlund operators, Cesaro, Riesz, Euler, Abel and many others. This research investigates and determines the spectrum of a class of N ¨ orlund operators on the sequence spaces c0, c and bv0. This is achieved by constructing the resolvent operator Tl = (T −Il )−1, the spectrum is then given by all the values of l for which Tl does not exist as a bounded operator on the sequence space c0, c and bv0. It is shown that the spectrum consists of the set {l ∈ C :

l − 1 3

≤ 1 3}∪{1}. This will find application in the development of Tauberian and Mercerian theorems for the No¨rlund operator which are used to determine the limit or sum of a convergent sequence or series. In addition the eigenvalues and the eigenvectors are used to solve infinite linear system of equations. Infinite dimensional linear systems appears naturally when studying control problems for systems modelled by linear partial differential equations. Many problems in dynamical systems can be written in form of infinite differential systems e.gMathieu equation, Hill’s equation.