Motivated by spectral properties of elliptic boundary value problems,
we fix K L(X,Y) compact and consider subsets G of $n(X,Y) with the
property that if T G, then T + AK is an isomorphism if A is large
and such that there is an orientation e which strongly orients G and
has the further property that
e(T) = degL
for A sufficiently large,
whenever T is an isomorphism in G. We show that if G Q $n(X,Y) is
convex and for each pair of isomorphisms T1 and T in G, there is some
A*0 with tT + (l-t)T +AK an isomorphism for t€[0,l] and AA#,
then G has the above properties. Moreover, G also has the above
property where G^. is defined to be the set of operators T $ (X,Y)
such that T + AK is an isomorphism for A large and, for fixed y Y,
(T+AK) (y) 0, in X, as A +00. In the case when X,W and F are
Banach spaces with X compactly embedded in W, if Y = W © T and
K(x) = (x,0) Y for x X, then an orientation e which strongly
orients G and has the property that e(T) = deg ((T+AK)" T) for A
sufficiently large, induces a degree which, for maps having principal parts
in G, has all of the properties of the Leray- Schauder degree. In
particular, it has the property that if f : X W © T is quasilinear
Fredholm and is C and f (*n) - 0 with f(xn) = (L ,B ) an isomorphism,
then if f(txQ) G for all t [0,1],
ind(f,xQ) = (-l)m
where m is the number of negative eigenvalues of
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