Compact Operators

Abstract

Let X and Y be complex B-spaces, and let S be the unit sphere in X. An operator T ∈ L (X, Y) is said to be compact or completely continuous if the image T · S is relatively compact in Y. For a compact operator T ∈ L(X, X), the eigenvalue problem can be treated fairly completely, in the sense that the classical theory of Fredholm concerning linear integral equations may be extended to the linear functional equation Tx - λx = ywith a complex parameter λ. This result is known as the Riesz-Schauder theory. F. Riesz [2] and J. Schauder [1].