On Compact Operators Whose Norms Are Eigenvalues and Completeness

Abstract

Let X be a Banach space and T: X→Y be a linear operator, then Tis compact if it maps bounded sequences in X to sequences in Y with convergent subsequences, that is, if xn ∈ X is a bounded sequence, then T xn ∈ Y has a convergent subsequence say, T xnk in Y. The eigenvalue of an operator T, is a scalar λ if there is a nontrivial solution x such that T x=λ x. Such an x is called an eigenvector corresponding to the eigenvalue λ. A vector space is complete if every Cauchy sequence in V converges in V. It is known that every finite dimensional nor medspace is complete and that a Hilbert space is a normed space that is complete with respect to the norm induced by the inner product. In this paper we have established the conditions for completeness of a compact operator T whose norm is an eigenvalue.