Adjoint is Involutive

Theorem

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

Let $\struct {\HH, \innerprod \cdot \cdot_\HH}$ and $\struct {\KK, \innerprod \cdot \cdot}$ be Hilbert spaces over $\mathbb F$.

Let $A : \HH \to \KK$ be a bounded linear transformation.

Define:

$A^{\ast \ast} = \paren {A^\ast}^\ast$

where $A^*$ denotes the adjoint of $A$.

Then $\ast$ is an involutive operation in the sense that:

$A^{\ast \ast} = A$

Proof

Let $x \in \HH$ and $y \in \KK$.

By the definition of the adjoint, we have:

$\innerprod {A y} x_\HH = \innerprod y {A^\ast x}_\KK$

Taking complex conjugates, we have:

$\overline {\innerprod {A y} x_\HH} = \overline {\innerprod y {A^\ast x}_\KK}$

Using the conjugate symmetry of the inner product, we have:

$\innerprod x {A y}_\HH = \innerprod {A^\ast x} y_\KK$

So by the definition of the adjoint:

$\innerprod x {A y}_\HH = \innerprod x {\paren {A^\ast}^\ast y}_\KK$

From the uniqueness of the adjoint, we obtain:

$\paren {A^\ast}^\ast = A$

$\blacksquare$

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $\text {II}.2.6 \ \text {(c)}$