Diagonal operator on sequence space

A diagonal operator on ${\ell }^p$ acts coordinatewise as $(x_n)\mapsto (a_n{x}_n)$. It is bounded when $(a_n)$ is bounded, while its inverse on its image is bounded only when the nonzero $a_n$ are bounded away from zero.

The injective diagonal map $(x_n)\mapsto (x_n/n)$ on ${\ell }^2$ is bounded, but its inverse on the range is unbounded; the range is correspondingly not closed.