Definition:Adjoint Linear Transformation

Definition
Let $H, K$ be Hilbert spaces.

Let $A \in \map B {H, K}$ be a bounded linear transformation.

By Existence and Uniqueness of Adjoint, there exists a unique bounded linear transformation $B \in \map B {K, H}$ such that:


 * ${\innerprod {A h} k}_K = {\innerprod h {B k} }_H$

We call $B$ the adjoint of $A$, and denote it $A^*$.

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