self-adjoint operator (original) (raw)

A densely defined linear operator A:𝒟⁢(A)⊂ℋ→ℋ on a Hilbert space ℋ is a Hermitian or symmetric operator if (A⁢x,y)=(x,A⁢y) for all x,y∈𝒟⁢(A). This means that the adjoint A* of A is defined at least on 𝒟⁢(A) and that its restriction to that set coincides with A. This fact is often denoted by A⊂A*.

The operator A is self-adjoint if it coincides with its adjoint, i.e. if A=A*. If A is closable and its closure coincides with its adjoint (i.e. A¯=A*), then A is said to be essentially self-adjoint.