nLab self-adjoint morphism

Definition

A morphism $f$ in a dagger-category is self-adjoint if

$$f^\dagger = f \,.$$

For appropriate choices of dagger categories this restricts to various notions of self-adjointness:

• a self-adjoint matrix is a self-adjoint morphism in the $\dagger$-category whose objects are $\mathbb{C}^n$s and whose morphisms are linear maps and the dagger-operation is transposition followed by complex conjugation;

• a bounded self-adjoint linear operator on a Hilbert space is defined everywhere and equals its own adjoint. The self-adjoint operator in general, unbounded case, is however not precisely equal to its adjoint as their domains may differ. As unbounded operators do not form an algebra this notion is nevertheless not a special case of the above. See self-adjoint operator.

Related concepts

• self-adjoint morphisms

• unitary morphism

• compact self-adjoint operator