Measure, Integration & Real Analysis, 2021a

304 Chapter 10 Linear Maps on Hilbert Spaces
An operator on a finite-dimensional vector space is left invertible if and only if
it is right invertible. We have seen that this result fails for bounded operators on
infinite-dimensional Hilbert spaces. However, the next result shows that we recover
this equivalency for normal operators.
10.55 invertibility for normal operators
Suppose V is a Hilbert space and T ∈B(V) is normal. Then the following are
equivalent:
(a) T is invertible.
(b) T is left invertible.
(c) T is right invertible.
(d) T is surjective.
(e) T is injective and has closed range.
(f)
T ∗ T is invertible.
(g) TT ∗ is invertible.
Proof Because T is normal, (f) and (g) are clearly equivalent. From 10.29, we know
that (f), (b), and (e) are equivalent to each other. From 10.31, we know that (g),
(c), and (d) are equivalent to each other. Thus (b), (c), (d), (e), (f), and (g) are all
equivalent to each other.
Clearly (a) implies (b).
Suppose (b) holds. We already know that (b) and (c) are equivalent; thus T is left
invertible and T is right invertible. Hence T is invertible, proving that (b) implies (a)
and completing the proof that (a) through (g) are all equivalent to each other.
The next result shows that a normal operator and its adjoint have the same eigenvectors,
with eigenvalues that are complex conjugates of each other. This result can
fail for operators that are not normal. For example, 0 is an eigenvalue of the left shift
on l 2 but its adjoint the right shift has no eigenvectors and no eigenvalues.
10.56 T normal and Tf = α f implies T ∗ f = α f
Suppose T is a normal operator on a Hilbert space V, α ∈ F, and f ∈ V. Then α
is an eigenvalue of T with eigenvector f if and only if α is an eigenvalue of T ∗
with eigenvector f .
Proof Because (T − αI) ∗ = T ∗ − αI and T is normal, T − αI commutes with its
adjoint. Thus T − αI is normal. Hence 10.53 implies that
‖(T − αI) f ‖ = ‖(T ∗ − αI) f ‖.
Thus (T − αI) f = 0 if and only if (T ∗ − αI) f = 0, as desired.
Measure, Integration & Real Analysis, by Sheldon Axler