Spectral Theory in Hilbert Space

CHAPTER 9
Extension theory
We will here complete the discussion on selfadjoint extensions of
a symmetric operator begun in Chapter 4. This material is originally
due to von Neumann although our proofs are different, and we will also
discuss an extension of von Neumann’s theory needed in Chapter 13.
1. Symmetric operators
We shall find criteria for the existence of selfadjoint extensions of a
densely defined symmetric operator, which according to the discussion
just before Example 4.4 must be a restriction of the adjoint operator.
We shall deal extensively with the graphs of various operators and it
will be convenient to use the same notation for the graph of an operator
T as for T itself. Note that if T is a closed operator on the Hilbert
space H, then its graph is a closed subspace of H ⊕ H, so in this case
T is itself a Hilbert space.
Recall that with the present notation we have
T ∗ = U(H ⊕ H) ⊖ T ) = (H ⊕ H) ⊖ UT
according to (4.1), where U : H ⊕ H ∋ (u, v) ↦→ (−iv, iu) is the boundary
operator introduced in Chapter 4. Also recall that U is selfadjoint,
unitary and involutary on H ⊕ H.
So, assume we have a densely defined symmetric operator T . We
want to investigate what selfadjoint extensions, if any, T has. Since
T ⊂ T ∗ the adjoint is densely defined and thus the closure T = T ∗∗
exists (Proposition 4.3) and is also symmetric. We may therefore as well
assume that T is closed to begin with. Recall that if S is a symmetric
extension of T , then it is a restriction of T ∗ since we then have T ⊂
S ⊂ S ∗ ⊂ T ∗ . Now put
D±i = {U ∈ T ∗ | UU = ±U}.
Note that U ∈ T ∗ means exactly that U = (u, T ∗ u) for some u ∈ D(T ∗ ).
It is immediately seen that Di and D−i consist of the elements of T ∗
of the form (u, iu) and (u, −iu) respectively, so that u satisfies the
equation T ∗ u = iu respectively T ∗ u = −iu. We may therefore identify
these spaces with the deficiency spaces D±i introduced in Chapter 5.
Also D±i are therefore called deficiency spaces.
Theorem 9.1 (von Neumann). If T is closed and symmetric operator,
then T ∗ = T ⊕ Di ⊕ D−i.
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