Spectral Properties of Compact Operators
Amonyela Hillary Isabu
Department of Mathematics, Masinde Muliro University of Science and Technology, P. O. BOX 190-50100, Kakamega, Kenya.
Achiles Nyongesa Simiyu *
Department of Mathematics, Masinde Muliro University of Science and Technology, P. O. BOX 190-50100, Kakamega, Kenya.
Aldrin Wekesa Wanambisi
Department of Mathematics, Masinde Muliro University of Science and Technology, P. O. BOX 190-50100, Kakamega, Kenya.
*Author to whom correspondence should be addressed.
Abstract
The spectral properties of a compact operator \(T : X \longrightarrow Y\) on a normed linear space resemble those of square matrices. For a compact operator, the spectral properties can be treated fairly completely in the sense that Fredholm's famous theory of integral equations may be extended to linear functional equations \(T x -\lambda\) \(= y\) with a complex parameter \(\lambda\) . This paper has studied and investigated the spectral properties of compact operators in Hilbert spaces. The spectral properties of compact linear operators are relatively simple generalization of the eigenvalues of finite matrices. As a result, the paper has given a number of corresponding propositions and interesting facts which are used to prove basic properties of compact operators. The Fredholm theory has been introduced to investigate the solvability of linear integral equations involving compact operators.
Keywords: Spectral properties, normed linear space, fredholm theory, compact operators