Just as the title. We know the image of $I-K$ is closed, but if we restrict $H$ to a closed subspace $V$, will $(I-K)(V)$ be a closed subspace of $H$? Any hint is appreciated.
2026-03-29 10:49:14.1774781354
$K$ is a linear compact operator on Hilbert space $H$. Will the image of $I-K$ on every closed subspace of $H$ be also closed?
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Hint: If $V\subset H$ is closed then, $I-K$ restricted to $V$, still being a proper map.