Spectral Theory in Hilbert Space

EXERCISES FOR CHAPTER 9 57
Exercises for Chapter 9
Exercise 9.1. Fill in all missing details in the proofs of Theorem
9.2 and Corollaries 9.3–9.5.
Exercise 9.2. Show that if n+ = n− < ∞, then if one selfadjoint
extension of a symmetric operator has compact resolvent, then every
other selfadjoint extension also has compact resolvent.
Hint: The difference of the resolvents for two selfadjoint extensions of
a symmetric operator has range contained in Dλ.
Exercise 9.3. Suppose T is a closed and symmetric operator on
H, that λ ∈ R and that Sλ is closed. Show that if λ is not an eigenvalue
of T , then T ∗ is the topological direct sum of T and Eλ and that
n+ = n−.
You may also show that if Sλ is closed but λ is an eigen-value of T ,
then one still has n+ = n−.
Exercise 9.4. Suppose T is a symmetric and positive operator,
i.e., 〈T u, u〉 ≥ 0 for every u ∈ D(T ). Use the previous exercise to show
that T has a selfadjoint extension (this is a theorem by von Neumann).
Exercise 9.5. Suppose T is a symmetric and positive operator. By
the previous exercise T has at least one selfadjoint extension. Prove
that there exists a positive selfadjoint extension (the so called Friedrichs
extension). This is a theorem by Friedrichs.
Hint: First define [u, v] = 〈T u, v〉 + 〈u, v〉 for u, v ∈ D(T ), show that
this is a scalar product, and let H1 be the completion of D(T ) in the
corresponding norm. Next show that H1 may be identified with a
subset of H and that for any u ∈ H the map H1 ∋ v ↦→ 〈v, u〉 is a
bounded linear form on H1. Conclude that 〈u, v〉 = [u, Gv] for u ∈ H1
and v ∈ H, where G is an operator on H with range in H1. Finally
show that G −1 − I, where I is the identity, is a positive selfadjoint
extension of T .
Exercise 9.6. Prove Theorem 9.11.
Exercise 9.7. Prove Theorem 9.12.
Exercise 9.8. Prove Corollaries 9.13–9.15.
Exercise 9.9. Prove Theorem 9.16.
Exercise 9.10. Prove Theorem 9.17.