Adjoint Operator

The adjoint $A^{†}:H\to H$ of an operator $A$ is defined by the relation $⟨Ax,y⟩=⟨x,A^{†}y⟩\forall x,y\in H$. Uniqueness and existence of $A^{\star }$ are assured by Riesz Representation Theorem.

where

• $H$ is an Hilbert Space
• $⟨\cdot ,\cdot ⟩$ is an Inner Product on $H$. We follow the Physics convention that an Inner Product is anti-linear ( linear ) on the 1st ( 2nd ) variable.
• $A$ is an Operator