A Generalization of the Riesz Theory of Completely Continuous Transformations
Lawrence M. Graves · Transactions of the American Mathematical Society · 1955
The classical theory, due to F. Riesz, of completely continuous transformations deals with a family Tc = I-cK of linear continuous transformations of a Banach space I into itself, where 7 is the identity and K is completely continuous.The transformation Te is then one-to-one, except for the "proper" values of c.In this paper we consider cases where the one-to-oneness no longer holds.The identity transformation 7 is replaced by a transformation E which maps 3£ onto the whole of another Banach space §), but not one-to-one.Then when K is completely continuous, the transformation Tc = E -cK maps £ onto g) except for a countable set of proper values ct which have no finite accumulation point.When 3Bi=rc£^g), c is said to be a proper value for A. We can no longer iterate the transformationTc, but we can apply Tc to A_ircX and obtain a linear closed space 2B2CSBi-After a finite number of repetitions of this process, we find a space 9B, = 9B,,+i.Then Tc transforms E-12B, onto 2J8" which takes the place of the invariant subspace of the Riesz theory.In the one-to-one case, there is associated with the transformation I -cK a resolvent Rc, which has a pole of order c at a proper value c0, and the Laurent expansion of Re about c0 leads to a decomposition of A into two mutually orthogonal parts.In the many-to-one case, the adjoint transformation T* is one-to-one except when c is a proper value, but the range of T* varies with c, so that it is not possible to define the order of a pole of the resolvent in the usual way.However a substitute definition has been found, as well as a decomposition K = G-\-H, where G has only one proper value Co,