Spectral Theory in Hilbert Space

52 9. EXTENSION THEORY
Proof. The facts that Di and D−i are eigenspaces of the unitary
operator U for different eigenvalues and 〈T, UT ∗ 〉 = 0 imply that T ,
Di and D−i are orthogonal subspaces of T ∗ (cf. Exercise 4.8). It
remains to show that Di ⊕ D−i contains T ∗ ⊖ T . However, U ∈ T ∗ ⊖ T
implies U ∈ H2 ⊖ T and thus UU ∈ T ∗ . Denoting the identity on
H2 by I and using U 2 = I one obtains U+ = 1
2 (I + U)U ∈ Di and
U− = 1
2 (I − U)U ∈ D−i. Clearly U = U+ + U− so this proves the
theorem.
We define the deficiency indices of T to be
n+ = dim Di = dim Di and n− = dim D−i = dim D−i
so these are natural numbers or ∞. We may now characterize the
symmetric extensions of T .
Theorem 9.2. If S is a closed, symmetric extension of the closed
symmetric operator T , then S = T ⊕ D where D is a subspace of
Di ⊕ D−i such that
D = {u + Ju | u ∈ D(J) ⊂ Di}
for some linear isometry J of a closed subspace D(J) of Di onto part of
D−i. Conversely, every such space D gives rise to a closed symmetric
extension S = T ⊕ D of T .
The proof is obvious after noting that if u+, v+ ∈ Di and u−, v− ∈
D−i, then 〈u+, v+〉 = 〈u−, v−〉 precisely if (u+ + u−, U(v+ + v−)) = 0.
Some immediate consequences of Theorem 9.2 are as follows.
Corollary 9.3. The closed symmetric operator T is maximal symmetric
precisely if one of n+ and n− equals zero and selfadjoint precisely
if n+ = n− = 0.
Corollary 9.4. If S is the symmetric extension of the closed symmetric
operator T given as in Theorem 9.2 by the isometry J with domain
D(J) ⊂ Di and range RJ ⊂ D−i, then the deficiency spaces for
S are Di(S) = Di ⊖ D(J) and D−i(S) = D−i ⊖ RJ respectively.
Proof. If D ⊂ Di ⊕ D−i and S = T ⊕ D is symmetric, then
u ∈ Di(S) ⊂ Di precisely if 〈T ⊕ D, Uu〉 = 0. But 〈T, Uu〉 = 0 and
if u+ + u− ∈ D with u+ ∈ Di, u− ∈ D−i then 〈u+ + u−, u〉 = 〈u+, u〉
which shows that Di(S) = Di ⊖ D(J). Similarly the statement about
D−i(S) follows.
Corollary 9.5. Every symmetric operator has a maximal symmetric
extension. If one of n+ and n− is finite, then all or none of the
maximal symmetric extensions are selfadjoint depending on whether
n+ = n− or not. If n+ = n− = ∞, however, some maximal symmetric
extensions are selfadjoint and some are not.