Definition:Adjoint Linear Transformation

Definition

Let $\HH$ and $\KK$ be Hilbert spaces.

Let $\map \BB {\HH, \KK}$ be the set of bounded linear transformations from $\HH$ to $\KK$.

Let $A \in \map \BB {\HH, \KK}$ be a bounded linear transformation.

By Existence and Uniqueness of Adjoint, there exists a unique bounded linear transformation $A^* \in \map \BB {\KK, \HH}$ such that:

$\forall h \in \HH, k \in \KK: {\innerprod {\map A h} k}_\KK = {\innerprod h {\map {A^*} k} }_\HH$

where $\innerprod \cdot \cdot_\HH$ and $\innerprod \cdot \cdot_\KK$ are inner products on $\HH$ and $\KK$ respectively.

$A^*$ is called the adjoint of $A$.

The operation of assigning $A^*$ to $A$ may be referred to as adjoining.

Also see

• Existence and Uniqueness of Adjoint, which ensures this concept is well-defined.

• Definition:Hermitian Operator
• Definition:Unitary Operator

• Results about adjoints can be found here.

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $\text {II}.2.4$