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 $λ_i∈R$.

**My question:** Is $T$ necessarily self-adjoint (i.e. $T=T^*$)? It is easy to see that $T$ is a symmetric operator and therefore $T \subset T^*$. I understand that I can make use of the definition of adjoint $T^*$ and $T$ to show that $T$ and $T^*$ agree on all basis vectors and therefore any finite linear combinations of them? But, how can one show that that they agree on the closure of the span? In other words, how do I show (if it is possible in the first place) that $T^* \subset T$?

Any help will be truly appreciated. Thanks.

definitelythink you should say that your question has been asked elsewhere, to avoid duplicating effort. For example, over at math.SE there is an excellent answer already... $\endgroup$