Compact Idempotent is of Finite Rank
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
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Sources
- 1990: John B. Conway: A Course in Functional Analysis ... (previous) ... (next): $II.4$: Exercise $4$
