ORIENTATION AND DEGREE FOR NONLINEAR BOUNDARY VALUE PROBLEMS 5
Our definition of degree turns upon first assigning a degree to each
linear isomorphism and then extending the degree to general quasilinear
Fredholm mappings. To motivate the first step, we recall the role of the
Brouwer and Leray-Schauder degrees in distinguishing connected components of
certain sets of linear isomorphisms.
If X and Y are of the same finite dimension, a choice of orienta-
tion of X and Y, respectively, defines the determinant, det (T), for
TeGL(X,Y). Then c : GL(X,Y)—{+l, -1}, defined by e(T) = sgn det (T),
distinguishes the two connected components of GL(X,Y). Of course, e(T)
is the Brouwer degree of T with respect to the choice of orientations.
If X is infinite dimensional, then GLp(X), the group of compact
vector fields in GL(X), also has two components, which are distinguished
by the function e : GLC(X) {-1,+1} defined by setting e(T) = (-l)n,
where n is the sum of the algebraic multiplicites of the negative eigen-
values of T. Of course, e(T) is the Leray-Schauder degree of T.
For general spaces X and Y, while GL(X,Y) may be connected, if we
divide GL(X,Y) into equivalence classes under the Calkin equivalence
relation, T ~ S if T-S is compact, then each equivalence class has two
connected components. It is reasonable to define the degree so that it
distinguishes the components of each Calkin equivalence class. Now, if T
and S in GL(X,Y) are equivalent, then they lie in the same component of
their equivalence class iff deg. ^ (T S) = 1. Accordingly, we define a
function e : GL(X,Y) {-1, +1} to be an orientation provided that if
T,S GL(X,Y) are equivalent, then
e(T)e(S) = degL
g
(T^S).
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