Definition:Adjoint Linear Transformation

Definition
Let $H, K$ be Hilbert spaces.

Let $A \in B \left({H, K}\right)$ be a bounded linear transformation.

Let $B \in B \left({K, H}\right)$ 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:Self-Adjoint Operator
 * Definition:Unitary Operator