Diagonalizable Operator Bounded iff Value Set Bounded

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Let $H$ be a Hilbert space.

Let $A: H \to H$ be a diagonalizable operator.

Let $\family {\alpha_e}_{e \mathop \in E}$ be the value set of $A$, with respect to a suitable basis $E$ for $H$.

Then $A$ is bounded if and only if $\family {\alpha_e}_{e \mathop \in E}$ is bounded.

That is, if and only if:

$\exists M \in \R: \forall e \in E: \cmod {\alpha_e} \le M$