Spectral Theory in Hilbert Space

13. FIRST ORDER SYSTEMS 97
a non-zero u ∈ ˜ P such that B(u, P) = 0. But then B is positive
definite on the linear hull of u and P, since B(αu + βv, αu + βv) =
|α| 2 B(u, u)+|β| 2 B(v, v) for any v ∈ P. This contradicts the maximality
of P as a positive definite space. From the standard fact dim ˜ P =
dim BP ( ˜ P) + dim{u ∈ ˜ P | BP u = 0} now follows that dim ˜ P ≤ dim P.
By symmetry all maximal positive definite subspaces for B have the
same dimension. Similarly, all maximal negative definite spaces for B
have the same dimension.
If P is any maximal positive definite subspace, and N any maximal
negative definite subspace, for B, we set r+ = dim P and r− = dim N .
The pair (r+, r−) is called the signature of the form B.
Proposition 13.11. Suppose P and N are maximal as positive
and negative definite subspaces for a Hermitian form B of finite rank.
Then P ∩ N = {0}, the direct sum P ˙+N is a maximal non-degenerate
space for B, and rank B = r+ + r−.
Proof. Clearly B can not be both positive and negative on the
same vector u, so P ∩M = {0}. B is obviously (check!) non-degenerate
on P ˙+N , and if P ˙+N is not maximal there exists u /∈ P ˙+N such that
B is non-degenerate on the linear hull M of u and P ˙+N . We may
assume B(u, P ˙+N ) = 0, since otherwise we can subtract from u its Bprojection
on P ˙+N . We cannot have B(u, u) = 0 since B would then
be degenerate on M. But if B(u, u) > 0, then B would be positive
definite on the linear hull of u and P, contradicting the maximality
of P. Similarly, if B(u, u) < 0 we would get a contradiction to the
maximality of N . Therefore P ˙+N is maximal non-degenerate so that
r+ + r− = rank B.
Two Hermitian forms Ba and Bb of finite rank are said to be independent
if each has a maximal non-degenerate space Ma respectively
Mb such that Ba(Mb, L) = Bb(Ma, L) = 0. It is then clear that
Ma ∩ Mb = {0} and that Ma ˙+Mb is maximal non-degenerate for
Bb − Ba. If (r a +, r a −) and (r b +, r b −) are the signatures of Ba and Bb respectively
it follows that (r b + + r a −, r b − + r a +) is the signature of Bb − Ba.
Now consider (13.3) and suppose I = (a, b). If u1 = (u1, v1) and
u2 = (u2, v2) ∈ T1 then −iu ∗ 2Ju1 has a limit both in a and b by (13.3).
We denote these limits Ba(u1, u2) and Bb(u1, u2) respectively and call
them the boundary forms at a and b respectively. Clearly Ba and Bb are
Hermitian forms on T1. Being limits of forms of rank n they both have
ranks ≤ n (Exercise 13.6). They are also independent. This follows
from the next lemma.
Lemma 13.12. Suppose (u, v) ∈ T1. Then there exists (u1, v1) in
T1 such that (u1, v1) = (u, v) in a right neighborhood of a and (u1, v1)
vanishes in a left neighborhood of b.