Spectral Theory in Hilbert Space

2. SYMMETRIC RELATIONS 55
then it is a restriction of T ∗ since we then have T ⊂ S ⊂ S ∗ ⊂ T ∗ .
Now put
D±i = {u ∈ T ∗ | Uu = ±u} .
It is immediately seen that Di and D−i consist of the elements of T ∗ of
the form (u, iu) and (u, −iu) respectively. We call them the deficiency
spaces of T . The following generalizes von Neumann’s formula.
Theorem 9.11. For any closed and symmetric relation T holds
T ∗ = T ⊕ Di ⊕ D−i.
The proof is the same as for Theorem 9.1 and is left to Exercise 9.6
As before 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 again natural numbers or ∞. The next theorem is completely
analogous to Theorem 9.2 with essentially the same proof, so
we leave this as Exercise 9.7
Theorem 9.12. If S is a closed, symmetric extension of the closed
symmetric relation 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 following consequences of Theorem 9.12 are completely analogous
to Corollaries 9.3–9.5, and their proofs are left as Exercise 9.8
Some immediate consequences of
Corollary 9.13. The closed symmetric relation T is maximal
symmetric precisely if one of n+ and n− equals zero and selfadjoint
precisely if n+ = n− = 0.
Corollary 9.14. If S is the symmetric extension of the closed
symmetric relation T given as in Theorem 9.12 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.
Corollary 9.15. Every symmetric relation 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.
We will now prove a theorem generalizing Theorem 9.6. To do this,
first note that Lemma 5.1 remains valid for relations, with the obvious