Kernel of an infinite dimensional operator

Question

Can the Kernel of an infinite dimensional operator have $\dim=0$? I am thinking to the annihilation ($\hat E$) and creation ($\hat E^\dagger$) operators. Suppose, in fact, we have an infinite but countable set of basis vectors $\{e_i\}_{i=0}^\infty$. We know that $\hat E^\dagger e_i=e_{i+1}$ and that $\hat Ee_i=e_{i-1}$ with $\hat Ee_0=0$. Does this means that while $\dim(\text{ker} \hat E)=1$, $\dim(\text{ker} \hat E^\dagger)=0$??