Diagonalizable Operator Compact iff Value Set Converges to Zero

Theorem

Let $H$ be a Hilbert space of countable dimension.

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

Let $\sequence {\alpha_n}_{n \mathop \in \N}$ be the value set of $A$, with respect to a suitable basis $E = \sequence {e_n}_{n \mathop \in \N}$ for $H$.

Then $A$ is compact if and only if:

$\ds \lim_{n \mathop \to \infty} \alpha_n = 0$

Proof

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Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $II.4.6$