Adjoint of Identity Transformation

Theorem

Let $\tuple {\HH, \innerprod \cdot \cdot_\HH}$ be a Hilbert space.

Let $I_\HH$ be the identity transformation on $\HH$.

Then:

${I_\HH}^* = I_\HH$

where ${I_\HH}^*$ denotes the adjoint of $I_\HH$.

Proof

From Identity Mapping on Normed Vector Space is Bounded Linear Operator:

$I_\HH$ is a bounded linear transformation.

So, from the existence part of Existence and Uniqueness of Adjoint:

$I_\HH$ has an adjoint ${I_\HH}^*$.

That is:

$\innerprod {I_\HH h} g_\HH = \innerprod h { {I_\HH}^* g}_\HH$

for all $h, g \in \HH$.

Note that by the definition of the identity transformation, we also have:

$\innerprod {I_\HH h} g_\HH = \innerprod h {I_\HH g}_\HH$

for all $h, g \in \HH$.

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

$I_\HH = {I_\HH}^*$

$\blacksquare$