# Self-adjoint

In mathematics, an element *x* of a *-algebra is **self-adjoint** if
.

A collection *C* of elements of a star-algebra is **self-adjoint** if it is closed under the involution operation. For example, if
then since
in a star-algebra, the set {*x*,*y*} is a self-adjoint set even though *x* and *y* need not be self-adjoint elements.

In functional analysis, a linear operator *A* on a Hilbert space is called **self-adjoint** if it is equal to its own adjoint *A*^{∗} and that the domain of *A* is the same as that of *A*^{∗}. See self-adjoint operator for a detailed discussion. If the Hilbert space is finite-dimensional and an orthonormal basis has been chosen, then the operator *A* is self-adjoint if and only if the matrix describing *A* with respect to this basis is Hermitian, i.e. if it is equal to its own conjugate transpose. Hermitian matrices are also called **self-adjoint**.

In a dagger category, a morphism
is called **self-adjoint** if
; this is possible only for an endomorphism
.

## References

- Reed, M.; Simon, B. (1972).
*Methods of Mathematical Physics*. Vol 2. Academic Press. - Teschl, G. (2009).
*Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators*. Providence: American Mathematical Society.