Norm of Adjoint

Theorem
Let $H, K$ be Hilbert spaces.

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

Then the norm of $A$ satisfies:


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

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

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

Then:

Therefore:

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 Double Adjoint is Itself, the reverse inequality is obtained.

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