ON THE EIGEN VALUES OF A NORLUND INFINITE MATRIX AS AN OPERATOR ON SOME SEQUENCE SPACES

Abstract

In various papers some authors have previously investigated and determined the spectrum of weighted mean matrices considered as bounded operators on various sequence spaces. It is evident that no much research has been done on the spectrum of Norlund matrices. In this study, we have investigated and determined the eigenvalues of a Norlund matrix as a bounded operator over the sequence spaces c0 and c. This was achieved by applying eigenvalue problem i.e Ax =l x. where l are numbers ( real or complex) and vector columns x( x 6= 0); such that x ∈ c0 and c. Also A∗x = l x such that x ∈ c∗0 and c∗ where c∗0 , c∗ = l1. The results obtained are A ∈ B(c0) , A ∈ B(bv0) and A ∈ B(l1) have no eigenvalues while the set of eigenvalus for A∗ ∈ B(l1) where c∗0 = l1 is {l ∈ C : |l +1| < 2}∪ {1}. Furthermore the set of eigenvalues for A ∈ B(c) is the singleton set {1} and that of A∗ ∈ B(l1) where c∗ = l1 is the set {l ∈ C : |l +1| < 2}∪{1} .The results from this research will provide useful information to engineers to improve on areas of application of eigenvalues and eigenvectors in engineering. It will also be useful to mathematicians when solving similar problems.