Hermitian Adjoint - Definition For Bounded Operators

Definition For Bounded Operators

Suppose H is a Hilbert space, with inner product . Consider a continuous linear operator A : H → H (this is the same as a bounded operator).

Using the Riesz representation theorem, one can show that there exists a unique continuous linear operator A* : H → H with the following property:

This operator A* is the adjoint of A. This can be seen as a generalization of the adjoint matrix of a square matrix which has a similar property involving the standard complex inner product.