Spectral Theory in Hilbert Space

2. SYMMETRIC RELATIONS 53
We will now generalize Theorem 9.1. To do this, we use the notation
of Lemma 5.1. Define
Dλ = {(u, λu) ∈ T ∗ } = {(u, λu) | u ∈ Dλ}
Eλ = {(u, λu + v) ∈ T ∗ | v ∈ D λ } .
It is clear that Eλ for non-real λ is the direct sum of Dλ and D λ since
if a = i
we have (u, λu + v) = a(v, λv) + (u − av, λ(u − av)). This
2 Im λ
direct sum is topological (i.e., the projections from Eλ onto Dλ and
Dλ are bounded) since all three spaces are obviously closed. Thus
the assertion follows from the closed graph theorem. Carry out the
argument as an exercise! We can now prove the following theorem.
Theorem 9.6. For any non-real λ we have T ∗ = T ˙+Eλ as a topological
direct sum.
Proof. Since all involved spaces are closed it is enough to show
the formula algebraically (the reason is as above). Let (u, v) ∈ T ∗ . By
Lemma 5.1.1 H = Sλ ⊕ D λ so we may write v − λu = w0 + w λ with
w λ ∈ D λ and w0 ∈ Sλ. We can find u0 ∈ H such that (u0, λuo+w0) ∈ T
so (u, v) = (u0, λu0 + w0) + (u − u0, λ(u − u0) + w λ ). The last term is
obviously in Eλ.
If (u, v) ∈ T ∩ Eλ we have v − λu ∈ Sλ ∩ D λ = {0} so that λ is an
eigenvalue of T if u = 0.
Corollary 9.7. If Im λ > 0 then dim Dλ = n+, dim D λ = n−.
Proof. Suppose U = (u, T ∗ u) and V = (v, T ∗ v) are in T ∗ . The
boundary form
〈U, UV 〉 = i(〈u, T ∗ v〉 − 〈T ∗ u, v〉)
is a bounded Hermitian form on T ∗ . It is immediately verified that
it is positive definite on Dλ, negative definite on D λ , non-positive on
T ˙+Dλ and non-negative on T ˙+D λ .
Let µ be a complex number with Im µ > 0. We get a linear map
of Dµ into Dλ in the following way. Given u ∈ Dµ we may write
u = u0 +uλ +u λ uniquely with u0 ∈ T , uλ ∈ Dλ and u λ ∈ D λ according
to Theorem 9.6. Let the image of u in Dλ be uλ. Then uλ can not be 0
unless u is since the boundary form is positive definite on Dµ but nonpositive
on T ˙+D λ . It follows that dim Dµ ≤ dim Dλ. By symmetry the
dimensions of Dλ and Dµ are then equal, i.e., dim Dλ = n+. Similarly
one shows that dim D λ = n−.
2. Symmetric relations
This section is a simplified version of Section 1 of [2]. Most of it can
also be found in [1]. The theory of symmetric and selfadjoint relations
is an easy extension of the corresponding theory for operators, but will
be essential for Chapters 13 and 14.