$T$ is invertible.

Question

Let $(e_j)$ an orthonormal basis of $L^2(\mathbb{R})$ and $T$ an unbounded operator self adjoint on $L^2(\mathbb{R})$ such that $\forall n\in \Bbb N ;||T(e_n)||>c||e_n||$ where $c>0$.

Can we say $T$ is invertible?.

Thank you in advance.