Norm of Adjoint

Theorem
Let $H, K$ be Hilbert spaces.

Let $A \in \map B {H, K}$ be a bounded linear transformation.

Let $A^* \in \map B {K, H}$ be the adjoint of $A$.

Then $A$ and $A^*$ satisfy:
 * $\norm A_{\map B {H, K} }^2 = \norm {A^*}_{\map B {K, H} }^2 = \norm {A^* A}_{\map B {H, H} }$

where:
 * $\norm \cdot_{\map B {H, K} }$ denotes the operator norm on $\map B {H, K}$
 * $\norm \cdot_{\map B {K, H} }$ denotes the operator norm on $\map B {K, H}$
 * $\norm \cdot_{\map B {H, H} }$ denotes the operator norm on $\map B {H, H}$

Proof
Let $h \in H$ such that $\norm h_H \le 1$.

Then:

By definition $(1)$ for $\norm A$, it follows that:
 * $\norm A^2 \le \norm {A^* A} \le \norm {A^*} \norm A$

That is:
 * $\norm A \le \norm {A^*}$.

By substituting $A^*$ for $A$, and using $A^{**} = A$ from Adjoint is Involutive, the reverse inequality is obtained.

Hence $\norm A^2 = \norm {A^* A} = \norm {A^*}^2$.