Question about domains of extensions of unbounded operators

Question

Let $T$ be a densely defined, symmetric operator on the Hilbert space $\mathcal{H}$ with the domain $D(T)$. Let $T_1$ and $T_2$ be self-adjoint extensions of $T$. Then obviously we have $$ T_1 f=T_2 f = T f, \qquad f\in D(T). $$ My question: is $T_1$ necessarily equal to $T_2$ outside of $D(T)$, whenever both $T_1$ and $T_2$ are defined. More precisely, is it true that $$ T_1 f=T_2 f, \qquad f\in D(T_1)\cap D(T_2)? $$ In other words, the question asks if self-adjoint extensions of a fixed symmetric operator differ only by their domains or can they actually have different values on some vectors? Any hints will be appreciated.

Answer 1

Since $T\subseteq T_1$ then $T_1={T_1}^*\subseteq T^*$. Similarly $T_2\subseteq T^*$. So if $v\in\mathcal{D}(T_1)\cap\mathcal{D}(T_2)$ then $T_1(v)=T_2(v)=T^*(v)$.