Measure, Integration & Real Analysis, 2021a

Section 10A Adjoints and Invertibility 281
10A
Adjoints and Invertibility
Adjoints of Linear Maps on Hilbert Spaces
The next definition provides a key tool for studying linear maps on Hilbert spaces.
10.1 Definition adjoint; T ∗
Suppose V and W are Hilbert spaces and T : V → W is a bounded linear map.
The adjoint of T is the function T ∗ : W → V such that
for every f ∈ V and every g ∈ W.
〈Tf, g〉 = 〈 f , T ∗ g〉
To see why the definition above makes
sense, fix g ∈ W. Consider the linear
functional on V defined by f ↦→ 〈Tf, g〉.
This linear functional is bounded because
The word adjoint has two unrelated
meanings in linear algebra. We need
only the meaning defined above.
|〈Tf, g〉|≤‖Tf‖‖g‖ ≤‖T‖‖g‖‖f ‖
for all f ∈ V; thus the linear functional f ↦→ 〈Tf, g〉 has norm at most ‖T‖‖g‖. By
the Riesz Representation Theorem (8.47), there exists a unique element of V (with
norm at most ‖T‖‖g‖) such that this linear functional is given by taking the inner
product with it. We call this unique element T ∗ g. In other words, T ∗ g is the unique
element of V such that
10.2 〈Tf, g〉 = 〈 f , T ∗ g〉
for every f ∈ V. Furthermore,
10.3 ‖T ∗ g‖≤‖T‖‖g‖.
In 10.2, notice that the inner product on the left is the inner product in W and the
inner product on the right is the inner product in V.
10.4 Example multiplication operators
Suppose (X, S, μ) is a measure space and h ∈L ∞ (μ). Define the multiplication
operator M h : L 2 (μ) → L 2 (μ) by
M h f = fh.
Then M h is a bounded linear map and ‖M h ‖≤‖h‖ ∞ . Because
∫
The complex conjugates that appear
〈M h f , g〉 = fhg dμ = 〈 f , M h
g〉 in this example are unnecessary (but
they do no harm) if F = R.
for all f , g ∈ L 2 (μ),wehaveM ∗ h = M h
.
Measure, Integration & Real Analysis, by Sheldon Axler