Definition:Adjoint Linear Transformation

Definition
Let $H, K$ be Hilbert spaces.

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

Let $B \in \map B {K, H}$ be the unique bounded linear transformation provided by Existence and Uniqueness of Adjoint.

Then $B$ is called the adjoint of $A$, and denoted $A^*$.

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