Measure, Integration & Real Analysis, 2021a

302 Chapter 10 Linear Maps on Hilbert Spaces
We showed that a bounded operator on a complex nonzero Hilbert space has a
nonempty spectrum. That result can fail on real Hilbert spaces (where by definition
the spectrum is contained in R). For example, the operator T on R 2 defined by
T(x, y) =(−y, x) has empty spectrum. However, the previous result and 10.38 can
be used to show that every self-adjoint operator on a nonzero real Hilbert space has
nonempty spectrum (see Exercise 9 for the details).
Although the spectrum of every self-adjoint operator is nonempty, it is not true that
every self-adjoint operator has an eigenvalue. For example, the self-adjoint operator
M x ∈B ( L 2 ([0, 1]) ) defined by (M x f )(x) =xf(x) has no eigenvalues.
Normal Operators
Now we consider another nice special class of operators.
10.50 Definition normal operator
A bounded operator T on a Hilbert space is called normal if it commutes with its
adjoint. In other words, T is normal if
T ∗ T = TT ∗ .
Clearly every self-adjoint operator is normal, but there exist normal operators that
are not self-adjoint, as shown in the next example.
10.51 Example normal operators
• Suppose μ is a positive measure, h ∈L ∞ (μ), and M h ∈B ( L 2 (μ) ) is the
multiplication operator defined by M h f = fh. Then M ∗ h = M h
, which means
that M h is self-adjoint if h is real valued. If F = C, then h can be complex
valued and M h is not necessarily self-adjoint. However,
M ∗ ∗
h M h = M |h| 2 = M h M h
and thus M h is a normal operator even when h is complex valued.
• Suppose T is the operator on F 2 whose matrix with respect to the standard basis
is ( )
2 −3
.
3 2
Then T is not self-adjoint because the matrix above is not equal to its conjugate
transpose. However, T ∗ T = 13I and TT ∗ = 13I, as you should verify. Because
T ∗ T = TT ∗ , we conclude that T is a normal operator.
10.52 Example an operator that is not normal
Suppose T is the right shift on l 2 ; thus T(a 1 , a 2 ,...)=(0, a 1 , a 2 ,...). Then T ∗
is the left shift: T ∗ (a 1 , a 2 ,...)=(a 2 , a 3 ,...). Hence T ∗ T is the identity operator
on l 2 and TT ∗ is the operator (a 1 , a 2 , a 3 ,...) ↦→ (0, a 2 , a 3 ,...). Thus T ∗ T ̸= TT ∗ ,
which means that T is not a normal operator.
Measure, Integration & Real Analysis, by Sheldon Axler