Answer by David Gao for Is the span of the eigenvectors of a non self-adjoint compact operator dense in the operator range?

This is not even true in finite dimensions. Say, the matrix $\begin{pmatrix} 1 & 1\\0 & 1 \end{pmatrix}$ has range the entire $\mathbb{C}^2$, but the span of eigenvectors is $\text{span}\{\begin{pmatrix} 1\\0 \end{pmatrix}\}$.