I would like to understand the following characterization of normal operators on a Hilbert space, for which I could not find a proof.
Let $T$ be a normal operator. Let $\lambda \in \mathbb{C}$, then $\lambda \in \textrm{Sp}(T)$ if and only if $\textrm{inf}_{x \in H} \{ \lVert Tx - \lambda x \rVert / \lVert x \rVert, x \neq 0 \}=0$.
What it is saying is that the spectrum consists entirely of approximate eigenvalues. This follows from the fact that for a normal operator the residual spectrum is empty.
Namely, the residual spectrum of $T$ are those $\lambda\in\sigma(T)$ which are not eigenvalues and such that $\def\ran{\operatorname{ran}}$ $\ran(T-\lambda I)$ is not dense. But if $\lambda$ is not an eigenvalue and $S=T-\lambda I$ then $\ker S=\{0\}$ and, since $S$ is normal, $$ \{0\}=\ker S=\ker S^*S=\ker SS^*=\ker S^*=\ran(S)^\perp, $$ which means that $\ran S=\ran(T-\lambda I)$ is dense.
With the residual spectrum being empty it follows that every $\lambda\in\sigma(T)$ is an approximate eigenvalue. For if $T-\lambda I$ has dense range and is not invertible, it cannot be bounded below.