Let $D: X\to Y$ and $L: Z\to Y$ are bounded linear operators between Banach spaces, and we further assume $D$ is Fredholm. Then can we conclude that the operator $$ D\oplus L: X\oplus Z \to Y$$ has a closed image?
2026-04-06 20:32:42.1775507562
The operator $D \oplus L$ with $D$ Fredholm has closed image?
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the image must contain that of $D.$ $D$'s image is of finite-codimension. Subspaces of finite co-dimension are closed so yes.
(to prove the last consider a mapping from the subspace plus a finite dim v-space that is bijective to the target: id on the subspace and mapping to a basis on the finite dim part.)