### 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.