Two important classes of linear operators on complex inner-product spaces are defined using the adjoint.

An operator $A$ is Hermitian or self-adjoint when $$A^\dagger=A.$$ For finite-dimensional matrices, Hermitian operators have real eigenvalues, and eigenvectors belonging to distinct eigenvalues are orthogonal.

For example, $$A=\begin{pmatrix}2&i\-i&3\end{pmatrix}$$ is Hermitian because taking the conjugate transpose returns the same matrix. A Hermitian operator is therefore the natural complex analogue of a real symmetric matrix.

An operator $U$ is unitary when $$U^\dagger U=UU^\dagger=I.$$ Unitary operators preserve inner products: $$\langle Uu,Uv\rangle=\langle u,v\rangle,$$ and therefore preserve norms. Their eigenvalues have magnitude one.

These two properties play different roles in quantum mechanics. Observables are represented by Hermitian operators so their eigenvalues—the possible ideal measurement values—are real. Closed-system changes of quantum state are unitary so normalization and inner products are preserved.

The distinction matters: Hermitian means 'equal to its adjoint' and is associated with real spectra; unitary means 'inverse equals adjoint' and is associated with norm preservation. An operator can have one property, both in special cases, or neither.