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

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

This operator is called the adjoint operator.

If the operator $$T$$ is a matrix, then $$T^*$$ is the Hermitian transpose (i.e., conjugate transpose) of $$T$$; that is, $$T^*=\bar{T}^T$$

Self-adjoint operators
If $$T = T^*$$, then $$T$$ is called a self-adjoint operator (D.4.5.4 ).

Unitary operators
If $$TT^* = T^*T  = I$$, then  $$T$$ is called unitary (D.4.5.4 ).

(Not so) special cases
If $$T$$ is a bounded linear operator, then (L4.5.2 )
 * 1) $$ (T^*)^* = T $$
 * $$||T || = || T^*|| $$