Normal element

In mathematics, an element "x" of a *-algebra is normal if it satisfies formula_1 
This definition stems from the definition of a normal linear operator in functional analysis, where a linear operator "A" from a Hilbert space into itself is called unitary if formula_2 where the adjoint of "A" is "A" and the domain of "A" is the same as that of "A". See normal operator for a detailed discussion. If the Hilbert space is finite-dimensional and an orthonormal basis has been chosen, then the operator "A" is normal if and only if the matrix describing "A" with respect to this basis is a normal matrix. 