Spectral Theory in Hilbert Space

54 9. EXTENSION THEORY
We call a (closed) linear subspace T of H 2 = H⊕H a (closed) linear
relation on H. This is a generalization of the concept of (the graph
of) a linear operator which will turn out to be useful in the following
chapters. We still denote by U the boundary operator on H 2 and define
the adjoint of the linear relation T on H by
T ∗ = H 2 ⊖ UT = U(H 2 ⊖ T ) .
Clearly T ∗ is a closed linear relation on H. Note that by not insisting
that T and T ∗ are graphs we can, for example, now deal with adjoints
of non-densely defined operators. Naturally T is called symmetric if
T ⊂ T ∗ and selfadjoint if T = T ∗ .
Proposition 9.8. Let T ⊂ S be linear relations on H. Then S ∗ ⊂
T ∗ . The closure of T is T = T ∗∗ and (T ) ∗ = T ∗ .
The reader should prove this proposition as an exercise. It is very
easy to obtain a spectral theorem for selfadjoint relations as a corollary
to the spectral theorem of Chapter 7. Given a relation T we call the set
D(T ) = {u ∈ H | (u, v) ∈ T for some v ∈ H} the domain of T . Now let
HT be the closure of D(T ) in H and put H∞ = {u ∈ H | (0, u) ∈ T ∗ }
One may view H∞ as the ‘eigen-space’ of T ∗ corresponding to the
‘eigen-value’ ∞.
Proposition 9.9. H = HT ⊕ H∞.
Proof. We have 〈(u, v), U(0, w)〉 = i〈u, w〉 so that (0, w) ∈ T ∗
precisely when w ∈ H ⊖ D(T ). The proposition follows.
Now assume T is selfadjoint and put T∞ = {0} × H∞ and ˜ T =
T ∩ H 2 T . Then it is clear that T = ˜ T ⊕ T∞ so we have split T into
its many-valued part T∞ and ˜ T which is called the operator part of T
because of the following theorem.
Theorem 9.10 (Spectral theorem for selfadjoint relations). If T is
selfadjoint, then ˜ T is the graph of a densely defined selfadjoint operator
in HT with domain D(T ).
Proof. ˜ T is the graph of a densely defined operator on HT since
(0, w) ∈ ˜ T implies w ∈ H∞ ∩ HT = {0}. ˜ T is selfadjoint since ˜ T =
T ⊖ T∞ so its adjoint (in HT ) is ˜ T ∗ = H 2 T ∩ (T ∗ ⊕ UT∞) = H 2 T ∩ T = ˜ T
(check this calculation carefully!).
It is now clear that we get a resolution of the identity for T by
adjoining the orthogonal projector onto H∞ to the resolution of the
identity for ˜ T .
Assume we have a symmetric relation T . We want to investigate
what selfadjoint extensions, if any, T has. Since the closure T of T is
also symmetric we may as well assume that T is closed to begin with.
Just as is the case for operators, if S is a symmetric extension of T ,