2. QUASILINEAR FREDHOLM MAPPINGS

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

m

of L(X,Y), on the connected components of which the Fredholm index is

constant.

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

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