Given a Hilbert space (separable) $\mathcal{H}$ with an orthonormal basis $\{e\_i\}\_{i=1}^{\infty}$, define an operator $T$ with domain $\mathcal{D}(T)$ equal to the span of $\{e\_i\}$ by $Te\_i:=\lambda\_ie\_i$, for some $λ\_i∈R$.

I am trying to show that $T$ is an essentially self-adjoint operator. That is, $T^\*=\overline{T}$. In order to do so, I only need to prove that $(T^{\*} \pm i)$ is injective. Or, equivalently, *ker*$(T^{\*} \pm i)=\{0\}.$

But I'm not entirely sure how to show that *ker*$(T^{\*} \pm i)=\{0\}.$

Any help or hint will be very useful. Thanks.