Adjoint of Composition of Linear Transformations is Composition of Adjoints

Theorem

Let $\mathbb F \in \set {\R, \C}$.

Let $\HH$, $\KK$ and $\LL$ be a Hilbert spaces over $\mathbb F$.

Let $A : \KK \to \LL$ and $B : \HH \to \KK$ be bounded linear transformations.

Then:

$\paren {A B}^* = B^* A^*$

where $^*$ denotes adjoining.

Proof

Let ${\innerprod \cdot \cdot}_\HH$, ${\innerprod \cdot \cdot}_\KK$ and ${\innerprod \cdot \cdot}_\LL$ denote inner products over $\HH$, $\KK$ and $\LL$ respectively.

Let $h \in \HH$ and $l \in \LL$.

Then:

\(\ds \innerprod {\map {\paren {A B} } h} l_\LL\)

\(=\)

\(\ds \innerprod {\map B h} {\map {A^*} l}_\KK\)

Definition of Adjoint Linear Transformation of $A$

\(\ds \)

\(=\)

\(\ds \innerprod h {\map {B^* A^*} l}_\HH\)

Definition of Adjoint Linear Transformation of $B$

We also have, by the definition of the adjoint:

$\innerprod {\map {\paren {A B} } h} l_\LL = \innerprod h {\map {\paren {A B}^*} l}_\LL$

So, by the uniqueness part of Existence and Uniqueness of Adjoint:

$\paren {A B}^* = B^* A^*$

$\blacksquare$

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $\text {II}.2.6 \ \text {(b)}$
• 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $13.1$: Existence of Hilbert Adjoint