Self-adjoint element - Wikipedia

Abstraction: Star-algebra element equal to its own adjoint

Key points:

• An element a of a -algebra is self-adjoint (Hermitian) if a = a; the set of self-adjoint elements forms a real linear subspace.
• For any element a, the combinations aa, aa, and the "real part" (a + a)/2 and "imaginary part" (a − a)/(2i) are always self-adjoint.
• In a C*-algebra, a normal element is self-adjoint if and only if its spectrum is a subset of the real numbers.
• Every self-adjoint element in a C*-algebra has a unique positive/negative part decomposition: a = a₊ − a₋ with a₊, a₋ ≥ 0 and a₊a₋ = 0.
• The continuous functional calculus maps real-valued continuous functions on the spectrum of a normal element to self-adjoint elements.

Connections: Self Adjoint · C Star Algebra · Spectral Theory · Operator Algebras

Source: http://en.wikipedia.org/wiki/Self-adjoint