Compact Idempotent is of Finite Rank

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Theorem

Let $H$ be a Hilbert space.

Let $T \in B_{0} \left({H}\right)$ be a compact linear operator, and let $T$ be idempotent.


Then $T \in B_{00} \left({H}\right)$, i.e., $T$ is a bounded finite rank operator.


Proof


Sources