18 PATRICK FITZPATRICK AND JACOBO PEJSACHOWICZ

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

g

((T+AK)""1!)

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

I

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