Self-adjoint

In mathematics, an element x of a *-algebra is self-adjoint if ${\displaystyle x^{*}=x}$.

A collection C of elements of a star-algebra is self-adjoint if it is closed under the involution operation. For example, if ${\displaystyle x^{*}=y}$ then since ${\displaystyle y^{*}=x^{**}=x}$ 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 ${\displaystyle f}$ is called self-adjoint if ${\displaystyle f=f^{\dagger }}$; this is possible only for an endomorphism ${\displaystyle f\colon A\to A}$.

See also

• Hermitian matrix
• Normal element
• Symmetric matrix
• Self-adjoint operator
• Unitary element

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.