Spectral Theory in Hilbert Space

4. OPERATORS 27
In the rest of this chapter we assume that H1 = H2 = H. A densely
defined operator T is then said to be symmetric if T ⊂ T ∗ . In other
words, if 〈T u, v〉 = 〈u, T v〉 for all u, v ∈ D(T ). Thus 〈T u, u〉 is always
real for a symmetric operator. It therefore makes sense to say that
a symmetric operator is positive if 〈T u, u〉 ≥ 0 for all u ∈ D(T ). A
densely defined symmetric operator is always closeable since T ∗ is automatically
densely defined, being an extension of T . If actually T = T ∗
the operator is said to be selfadjoint. This is an important property
because these are the operators for which we will prove the spectral
theorem. In practice it is usually quite easy to see if an operator is
symmetric, but much more difficult to decide whether a symmetric operator
is selfadjoint. When one wants to interpret a differential operator
as a Hilbert space operator one has to choose a domain of definition;
in many cases it is clear how one may choose a dense domain so that
the operator becomes symmetric. With luck this operator may have
a selfadjoint closure 3 , in which case the operator is said to be essentially
selfadjoint. Otherwise, given a symmetric T , one will look for
selfadjoint extensions of T . If S is a symmetric extension of T , we get
T ⊂ S ⊂ S ∗ ⊂ T ∗ so that any selfadjoint extension of T is a restriction
of the adjoint T ∗ . There is now obviously a need for a theory of
symmetric extensions of a symmetric operator. We will postpone the
discussion of this until Chapter 9. Right now we will instead study
some very simple, but typical, examples.
Example 4.4. Consider the differential operator d
on some open
dx
interval I. We want to interpret it as a densely defined operator in
the Hilbert space L2 (I) and so must choose a suitable domain. A convenient
choice, which would work for any differential operator with
smooth coefficients, is the set C∞ 0 (I) of infinitely differentiable functions
on I with compact support, i.e., each function is 0 outside some
compact subset of I. It is well known that C∞ 0 (I) is dense in L2 (I).
Let us denote the corresponding operator T0; it is usually called the
minimal operator for d . Sometimes it is the closure of this operator
dx
which is called the minimal operator, but this will make no difference
to the calculations in the sequel. We now need to calculate the adjoint
of the minimal operator.
Let v ∈ D(T ∗ 0 ). This means that there is an element v∗ ∈ L2 (I)
such that
I ϕ′ v =
I ϕv∗ for all ϕ ∈ C∞ 0 (I) and that T ∗ 0 v = v∗ . In-
tegrating by parts we have
I ϕv∗ = −
I (ϕ′ v∗ ) since the boundary
terms vanish. Here v∗ denotes any integral function of v∗ . Thus we
have
I ϕ′ (v + v∗ ) = 0 for all ϕ ∈ C∞ 0 (I). We need the following
lemma.
3 This is the same as T ∗ being selfadjoint. Show this!