We first establish notation and recall some results regarding linear
Fredholm operators (see [G-G-K]).
Let X and Y be real Banach spaces. By L(X,Y) we denote the
space of bounded linear operators from X to Y, and consider L(X,Y) as
a Banach space with the usual norm topology. By GL(X,Y) we denote the
set of invertible operators in L(X,Y), and by K(X,Y) we denote the set
of compact operators in L(X,Y). When X = Y we set L(X) = L(X,Y),
L (X) = {T = Id - K|K K(X)} and GLC(X) = LC(X) n GL(X).
An operator T in L(X,Y) is called Fredholm if the kernel of T,
Ker T, has finite dimension and the image of T, Im T, is of finite
codimension in Y. We denote the set of Fredholm operators by $(X,Y).
For T $(X,Y), the numerical Fredholm index of T, Ind(T), is
defined by
Ind(T) = dimCKer T) - codimClm T),
and for m an integer the subset of L(X,Y) consisting of Fredholm
operators of index m is denoted by $ (X,Y). $(X,Y) is an open subset
of L(X,Y), on the connected components of which the Fredholm index is
Let T $ (X,Y). If we choose Q L(X) to be a projection onto
Ker T and let S : Ker T -^ Coker T be a linear bisection, then K = S o Q
is compact. Moreover, M = T + K€GL(X,Y) and if R = M_1, then
RT- Id€K(X).
According to the classical Riesz-Schauder Theorem, each linear compact
perturbation of the identity is Fredholm of index 0. More generally, a
Previous Page Next Page