Adjoint of Symmetric Densely-Defined Linear Operator Extends Operator

Theorem

Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space.

Let $\struct {\map D {T^\ast}, T^\ast}$ be the adjoint of $T$.

Then $\map D T \subseteq \map D {T^\ast}$ and:

$T x = T^\ast x$ for each $x \in \map D T$.

For each $y \in \HH$, define the linear functional $f_x : \map D T \to \Bbb F$ by:

$\map {f_y} x = \innerprod {T x} y$ for each $x \in \map D T$.

We show that for $y \in \map D T$, $f_y$ is bounded.

Let $y \in \map D T$, then:

$\innerprod {T x} y = \innerprod x {T y}$ for each $x \in \map D T$.

Then we have:

$\ds \cmod {\map {f_y} x}$

$=$

$\ds \cmod {\innerprod x {T y} }$

$\ds $

$\le$

$\ds \norm {T y} \norm x$

So $f_y$ is bounded and $\map D T \subseteq \map D {T^\ast}$.

Now, for $x, y \in \map D T$, we have:

$\innerprod {T x} y = \innerprod x {T y}$

since $\struct {\map D T, T}$ is symmetric and:

$\innerprod {T x} y = \innerprod x {T^\ast y}$

from the definition of the adjoint.

So we have:

$\innerprod x {T y - T^\ast y} = 0$

for each $x \in \map D T$.

Taking a sequence $\sequence {x_n}_{n \mathop \in \N}$ in $\map D T$ such that $x_n \to T y - T^\ast y$.

Since:

$\innerprod {x_n} {T y - T^\ast y} = 0$

for each $n \in \N$, we have:

$\innerprod {T y - T^\ast y} {T y - T^\ast y} = \norm {T y - T^\ast y}^2 = 0$

So $T y = T^\ast y$ for each $y \in \map D T$.

$\blacksquare$