Measure, Integration & Real Analysis, 2021a

Section 10A Adjoints and Invertibility 291
Now suppose (c) holds, so T is injective and has closed range. We want to prove
that (a) holds. Let R : range T → V be the inverse of the one-to-one linear function
f ↦→ Tf that maps V onto range T. Because range T is a closed subspace of V and
thus is a Banach space [by 6.16(b)], the Bounded Inverse Theorem (6.83) implies
that R is a bounded linear map. Let P denote the orthogonal projection of V onto the
closed subspace range T. Define S : V → V by
Then for each g ∈ V,wehave
Sg = R(Pg).
‖Sg‖ = ‖R(Pg)‖ ≤‖R‖‖Pg‖ ≤‖R‖‖g‖,
where the last inequality comes from 8.37(d). The inequality above implies that S is
a bounded operator on V. Iff ∈ V, then
S(Tf)=R ( P(Tf) ) = R(Tf)= f .
Thus ST = I, which means that T is left invertible, completing the proof that (c)
implies (a).
At this stage of the proof we know that (a), (b), and (c) are equivalent. To prove
that one of these implies (d), suppose (b) holds. Squaring the inequality in (b), we
see that if f ∈ V, then
‖ f ‖ 2 ≤ α 2 ‖Tf‖ 2 = α 2 〈T ∗ Tf, f 〉≤α 2 ‖T ∗ Tf‖‖f ‖,
which implies that
‖ f ‖≤α 2 ‖T ∗ Tf‖.
In other words, (b) holds with T replaced by T ∗ T (and α replaced by α 2 ). By the
equivalence we already proved between (a) and (b), we conclude that T ∗ T is left
invertible. Thus there exists S ∈B(V) such that S(T ∗ T)=I. Taking adjoints of
both sides of the last equation shows that (T ∗ T)S ∗ = I. Thus T ∗ T is also right
invertible, which implies that T ∗ T is invertible. Thus (b) implies (d).
Finally, suppose (d) holds, so T ∗ T is invertible. Hence there exists S ∈B(V)
such that I = S(T ∗ T)=(ST ∗ )T. Thus T is left invertible, showing that (d) implies
(a), completing the proof that (a), (b), (c), and (d) are equivalent.
You may be familiar with the finite-dimensional result that right invertibility is
equivalent to surjectivity. The next result shows that this equivalency also holds on
infinite-dimensional Hilbert spaces.
10.31 right invertibility
Suppose V is a Hilbert space and T ∈B(V). Then the following are equivalent:
(a) T is right invertible.
(b) T is surjective.
(c) TT ∗ is invertible.
Measure, Integration & Real Analysis, by Sheldon Axler