Restriction of an isomorphism to an invariant subspace may fail to be surjective 
I'm wondering whether the restriction of a vector space automorphism $f : V \to V$ to an invariant subspace $W \subset V$ can fail to be surjective, i.e. $f\vert_W : W \to W$ is not an automorphism.

Clearly, this can only happen if $W$ (and also $V$) is infinite dimensional, since $f\vert_W$ is injective.
I tried to look at $V=l^1(\Bbb N)$ and 
$f : (x_1,x_2,x_3, x_4, \dots) \mapsto (x_2,x_1,x_4, x_3, \dots) \in \text{GL}(V)$
or at some $f$ so that $f\vert_W : (x_1,x_2,x_3, x_4, \dots) \mapsto (0,x_1,x_2,x_3, x_4, \dots)$. But I wasn't able to conclude.
Thank you for your help!

Try a shift operator on $\ell^1(\mathbb Z)$.
