Let $H$ be a Hilbert space and suppose we have a bounded closed range operator $T\in B(H)$. Is $T(M)$ closed for every closed subspace $M\subset H$ ?
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2No! This is not even true if $T$ is an orthogonal projection. See here: https://math.stackexchange.com/questions/163844/is-an-open-linear-map-closed-to-some-extent However, if the kernel is finite-dimensional (i.e., $T$ is upper semi-Fredholm), then the answer is always yes. – amsmath Oct 11 '19 at 16:08
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Thank you so much – Djalal Ounadjela Oct 11 '19 at 16:36