Spectral Theory in Hilbert Space

24 4. OPERATORS
(5) T ∗ = T ,
(6) T ∗ T = T 2 .
Proof. The first four properties are very easy to show and are
left as exercises for the reader. To prove (5), note that we already
have shown that T ∗ ≤ T and combining this with (4) gives the
opposite inequality. Use of (5) shows that T ∗ T ≤ T ∗ T =
T 2 and the opposite inequality follows from T u 2 2 = 〈T ∗ T u, u〉1 ≤
T ∗ T u1u1 ≤ T ∗ T u 2 1 so (6) follows. The reader is asked to fill
in the details missing in the proof (Exercise 4.3).
If H1 = H2 = H3 = H, then the properties (1)–(4) above are the
properties required for the star operation to be called an involution
on the algebra B(H), and a Banach algebra with an involution, also
satisfying (5) and (6), is called a B ∗ algebra. There are no less than
three different useful notions of convergence for operators in B(H1, H2).
We say that Tj tends to T
• uniformly if Tj − T → 0, denoted by Tj ⇒ T ,
• strongly if Tju−T u2 → 0 for every u ∈ H1, denoted Tj → T ,
and
• weakly if 〈Tju, v〉2 → 〈T u, v〉2 for all u ∈ H1 and v ∈ H2,
denoted Tj ⇀ T .
It is clear that uniform convergence implies strong convergence and
strong convergence implies weak convergence, and it is also easy to see
that neither of these implications can be reversed.
Of particular interest are so called projection operators. A projection
P on H is an operator in B(H) for which P 2 = P . If P is
a projection then so is I − P , where I is the identity on H, since
(I − P )(I − P ) = I − P − P + P 2 = I − P . Setting M = P H and
N = (I − P )H it follows that M is the null-space of I − P since M
clearly consist of those elements u ∈ H for which P u = u. Similarly
N is the null-space of P . Since P and I − P are bounded (i.e., continuous)
it therefore follows that M and N are closed. It also follows
that M ∩ N = {0} and the direct sum M ˙+N of M and N is H (this
means that any element of H can be written uniquely as u + v with
u ∈ M and v ∈ N). Conversely, if M and N are linear subspaces of H,
M ∩ N = {0} and M ˙+N = H, then we may define a linear map P satisfying
P 2 = P by setting P w = u if w = u +v with u ∈ M and v ∈ N.
As we have seen P can not be bounded unless M and N are closed.
There is a converse to this: If M and N are closed, then P is bounded.
This follows immediately from the closed graph theorem (Exercise 4.4).
In the case when the projection P , and thus also I − P , is bounded,
the direct sum M ∔ N is called topological. If M and N happen to be
orthogonal subspaces P is called an orthogonal projection. Obviously
N = M ⊥ then, since the direct sum of M and N is all of H. We have
the following characterization of orthogonal projections.