Self-adjoint

In mathematics, and more specifically in abstract algebra, an element x of a *-algebra is self-adjoint if $x^{*}=x$ . A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold.

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

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.

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Self-adjoint

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