Friedrichs extension

Remarks

The operator $T_F$ is a self-adjoint extension of $T$, and $\ip{T_F x}{x} \ge c \norm{x}^2$ for all $x \in D(T_F)$. An equivalent description is that $T_F$ is the restriction of the adjoint $T^*$ to $D(T^*) \cap H_T$. The space $H_T$ is the form domain of $T_F$, and the sesquilinear form $(x, y) \mapsto \ip{Tx}{y}$ extends to it.

Example: on $H = L^2(0, 1)$, let $T f = -f''$ on smooth functions with compact support in $(0, 1)$. Integration by parts gives $\ip{Tf}{f} = \int_0^1 \abs{f'}^2 \, dx \ge 0$, so $c = 0$ works. The space $H_T$ is the Sobolev space $H^1_0(0, 1)$, and $T_F f = -f''$ on $D(T_F) = H^2(0, 1) \cap H^1_0(0, 1)$, the Dirichlet Laplacian.

Non-example: on $L^2(0, \infty)$, let $S f = -i f'$ on smooth functions with compact support in $(0, \infty)$. It is symmetric, but $\ip{Sf}{f}$ is not bounded below by any $c \norm{f}^2$, so the construction does not apply. In fact the deficiency indices of $S$ are unequal, so $S$ has no self-adjoint extension at all.