Norm of Adjoint

Theorem
Let $H, K$ be Hilbert spaces.

Let $A \in B \left({H, K}\right)$ be a bounded linear transformation.

Then the norm of $A$ satisfies:


 * $\left\Vert{A}\right\Vert^2 = \left\Vert{A^*}\right\Vert^2 = \left\Vert{A^*A}\right\Vert$

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

Proof
Let $h \in H$ such that $\left\Vert{h}\right\Vert_H \le 1$.

Then:

By definition $(1)$ for $\left\Vert{A}\right\Vert$, it follows that $\left\Vert{A}\right\Vert^2 \le \left\Vert{A^*A}\right\Vert \le \left\Vert{A^*}\right\Vert \left\Vert{A}\right\Vert$.

That is, $\left\Vert{A}\right\Vert \le \left\Vert{A^*}\right\Vert$.

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

Hence $\left\Vert{A}\right\Vert^2 = \left\Vert{A^*A}\right\Vert = \left\Vert{A^*}\right\Vert^2$.