Adjoint

Under a Hilbert space $$\mathcal H$$, an adjoint operator is defined in terms of inner product. Let $$T$$ be a bounded linear operator. Then, there exists (T.4.4.3 ) an operator $$T^*$$ such that

$$\langle T\mathbf{v}, \mathbf{w}\rangle_{\mathcal H} = \langle \mathbf{v}, T^*\mathbf{w} \rangle.$$

This operator is called the adjoint operator.

Self-adjoint operators
If $$T = T^*$$, then $$T$$ is called a self-adjoint operator.