normal operator equation

Question

let $S: V \to V$ linear transformation in a inner product space of $\mathbb{C}$.

Prove that $S$ is normal iff $$\|S(x)\| = \|S^*(x)\|$$

That's what I have done so far:

if $S$ is normal than $$SS^* = S^*S \implies \|S(X)\|^2 = (S(x),S(x)) = (x, S^*S(x)) = (x,SS^*(x)) = (S^*(x), S^*(x)) = \|S^*(x)\|^2$$

Thats for the first part, how do I prove that if $\|S(x)\| = \|S^*(x)\|$ than $S$ is normal? ($SS^* = S^*S$)

Answer 1

Using the polarization identity, show that if you have two linear maps $T,R \colon V \rightarrow V$ such that $\left< Tv, v \right> = \left< Rv, v \right>$ for all $v \in V$ then $T = R$. Apply this to $T = SS^{*}$ and $R = S^{*}S$.