Self-adjoint

In mathematics, an element "x" of a *-algebra is self-adjoint if formula_1.
A collection "C" of elements of a star-algebra is self-adjoint if it is closed under the involution operation. For example, if formula_2 then since formula_3 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 formula_4 is called self-adjoint if formula_5; this is possible only for an endomorphism formula_6.