Finite-rank operator

A finite-rank operator has finite-dimensional image and is compact.

For a finite-rank operator $S$ and $\lambda \ne 0$, if $\lambda$ is not an eigenvalue then $S-\lambda I$ is surjective. Decompose the Hilbert space into $\mathrm{im}S$ and its orthogonal complement, then use injectivity and surjectivity equivalence on the finite-dimensional image.