Definition:Adjoint of Densely-Defined Linear Operator

Definition

Let $\Bbb F \in \set {\R, \C}$.

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

Let $\struct {\map D T, T}$ be a densely defined linear operator on $\HH$.

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 $x \in \map D T$.

Define:

$\map D {T^\ast} = \set {y \in H : f_y \text { is a bounded linear functional}}$

We define the adjoint of $\struct {\map D T, T}$, $\struct {\map D {T^\ast}, T^\ast}$ as the unique linear transformation $T^\ast : \map D {T^\ast} \to \HH$ with:

$\innerprod {T x} y = \innerprod x {T^\ast y}$ for all $x \in \map D T$ and $y \in \map D {T^\ast}$.

Note that although we adopt the notation $\struct {\map D {T^\ast}, T^\ast}$ for convenience, which mirrors that of a densely-defined linear operator, $\map D {T^\ast}$ may not be everywhere dense.

An example is exhibited in Adjoint of Densely-Defined Linear Operator may not be Densely-Defined.

Also see

• Existence and Uniqueness of Adjoint of Densely-Defined Linear Operator

Sources

• 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $25.1$: Adjoints of Unbounded Operators