Spectral Theory in Hilbert Space

4. OPERATORS 25
Proposition 4.2. A projection P is orthogonal if and only if it
satisfies P ∗ = P .
Proof. If P ∗ = P and u ∈ M, v ∈ N, then 〈u, v〉 = 〈P u, v〉 =
〈u, P ∗ v〉 = 〈u, P v〉 = 〈u, 0〉 = 0 so M and N are orthogonal. Conversely,
suppose M and N orthogonal. For arbitrary u, v ∈ H we
then have 〈P u, v〉 = 〈P u, P v〉 + 〈P u, (I − P )v〉 = 〈P u, P v〉 so that
also 〈u, P v〉 = 〈P u, P v〉. Hence 〈P u, v〉 = 〈u, P v〉 holds generally, i.e.,
P ∗ = P .
An operator T for which T ∗ = T is called selfadjoint. Hence an
orthogonal projection is the same as a selfadjoint projection. We will
have much more to say about selfadjoint operators in a more general
context later. Another class of operators of great interest are the unitary
operators. This is an operator U : H1 → H2 for which U ∗ = U −1 .
Since 〈Uu, Uv〉2 = 〈U ∗ Uu, v〉1 = 〈u, v〉1 the operator U preserves the
scalar product; such an operator is called isometric. If U is isometric
we have 〈u, v〉1 = 〈Uu, Uv〉2 = 〈U ∗ Uu, v〉1, so that U ∗ is a left
inverse of U for any isometric operator. If dim H1 = dim H2 < ∞,
then a left inverse of a linear operator is also a right inverse, so in
this case isometric and unitary (orthogonal in the case of a real space)
are the same thing. If dim H1 = dim H2 or both spaces are infinitedimensional,
however, this is not the case. For example, in the space
ℓ 2 we may define U(x1, x2, . . . ) = (0, x1, x2, . . . ), which is obviously
isometric (this is a so called shift operator), but the vector (1, 0, 0, . . . )
is not the image of anything, so the operator is not unitary. Its adjoint
is U ∗ (x1, x2, . . . ) = (x2, x3, . . . ), which is only a partial isometry,
namely an isometry on the vectors orthogonal to (1, 0, 0, . . . ). See also
Exercise 4.8.
It is never possible to interpret a differential operator as a bounded
operator on some Hilbert space of functions. We therefore need to
discuss unbounded operators as well. Similarly, we will need to discuss
operators that are not defined on all of H. Thus we now consider a
linear operator T : D(T ) → H2, where the domain D(T ) of T is some
linear subset of H1. T is not supposed bounded. Another such operator
S is said to be an extension of T if D(T ) ⊂ D(S) and Su = T u for
every u ∈ D(T ). We then write T ⊂ S. We must discuss the concept
of adjoint. The form u ↦→ 〈T u, v〉2 is, for fixed v ∈ H2, only defined for
u ∈ D(T ), and though linear not necessarily bounded, so there may not
be any v ∗ ∈ H1 such that 〈T u, v〉2 = 〈u, v ∗ 〉1 for all u ∈ D(T ). Even
if there is, it may not be uniquely determined, since if w ∈ D(T ) ⊥ we
could replace v ∗ by v ∗ + w with no change in 〈u, v ∗ 〉. We therefore
make the basic assumption that D(T ) ⊥ = {0}, i.e., D(T ) is dense in
H1. T is then said to be densely defined 2 . In this case v ∗ ∈ H1 is
ter 9.
2 We will discuss the case of an operator which is not densely defined in Chap