Another definition of normal operator

Question

Another definition of a normal operator is that $T$ is normal if and only if $\| \textsf{T}v \| = \| \textsf{T}^∗ v \|$ for all $v$.

Can someone please explain to me the definition and possibly give me some intuition for why this definition makes sense? And how it connects to the usual definition of a normal operator?

Answer 1

Let $H$ be a Hilbert space. A continuous linear operator $T: H \rightarrow H$ is normal iff $T^* T = TT^*$. Note that: $$ \| Tx \|^2 = \langle T^* T x, x \rangle = \langle T T^* x, x \rangle = \|T^*x \|^2 $$ for all $x \in H$. From this, the equivalence follows. To see why the first equality above is true, recall that: $$ \langle T x, x \rangle = \langle x, T^* x \rangle $$ by definition of adjoint operator.

Answer 2

If you assume that $\|Tv\| = \|T^\ast v\|$ holds for all $v$, the equalities

$$\|Tv\|^2 = \langle Tv, Tv \rangle = \langle T^\ast Tv, v \rangle = \langle TT^\ast v, v \rangle = \langle T^\ast v, T^\ast v \rangle = \| T^\ast v \|^2$$

are true. But from $\langle T^\ast Tv, v \rangle = \langle TT^\ast v, v \rangle $ for all $v$ it follows that $T^\ast T = TT^\ast$.