Spectral Theory in Hilbert Space

4. OPERATORS 29
must both be 0. Hence 〈T1u, v〉 − 〈u, T1v〉 = 0 for any u, v ∈ D(T1), so
the maximal operator is symmetric and therefore selfadjoint (how does
this follow?). It also follows that the maximal operator is the closure of
the minimal operator so the minimal operator is essentially selfadjoint.
Example 4.7. Consider the same operator as in Example 4.6 but
for the interval (0, ∞). If u ∈ D(T1) we obtain 〈T1u, u〉 − 〈u, T1u〉 =
i|u(0)| 2 . To have a symmetric restriction of T1 we must therefore require
u(0) = 0, and with this restriction on the domain of T1 we obtain a
maximal symmetric operator T . If now u ∈ D(T ) and v ∈ D(T1) we
obtain 〈T u, v〉−〈u, T1v〉 = iu(0)v(0) = 0 so that T ∗ = T1. T is therefore
not selfadjoint so no matter how we choose the domain the differential
, though formally symmetric, will not be selfadjoint in
operator −i d
dx
L2 (0, ∞). One says that −i d
dx has no selfadjoint realization in L2 (0, ∞).
Example 4.8. We finally consider the operator of Example 4.6 for
the interval (−π, π). We now have
(4.2) 〈T1u, v〉 − 〈u, T1v〉 = −i(u(π)v(π) − u(−π)v(−π)).
In particular, for u = v it follows that for u to be in the domain of a
symmetric restriction of T1 we must require |u(π)| = |u(−π)| so that u
satisfies the boundary condition u(π) = e iθ u(−π) for some real θ. From
(4.2) then follows that if v is in the domain of the adjoint, then v will
have to satisfy the same boundary condition. On the other hand, if we
impose this condition, then the resulting operator will be selfadjoint
(because its adjoint will be symmetric). It follows that restricting the
domain of T1 by such a boundary condition is exactly what is required
to obtain a selfadjoint restriction. Each θ in [0, 2π) gives a different
selfadjoint realization, but there are no others.
The examples show that there may be a unique selfadjoint realization
of our formally symmetric differential operator, none at all, or
infinitely many depending on circumstances. It can be a very difficult
problem to decide which of these possibilities occur in a given case.
In particular, much effort has been devoted to decide whether a given
differential operator on a given domain has a unique selfadjoint realization.