Selfadjointness

Self-adjointness is a property of linear operators on inner product spaces, most often on complex Hilbert spaces. An operator A with domain D(A) is self-adjoint if it equals its adjoint A*, meaning D(A) = D(A*) and <Ax, y> = <x, Ay> for all x, y in D(A). In finite dimensions this is equivalent to the matrix representing A being equal to its conjugate transpose.

For unbounded operators, the adjoint is defined via the inner product and requires careful domain considerations. A symmetric (or Hermitian) operator satisfies <Ax, y> = <x, Ay> for all x, y in D(A), but need not have D(A) = D(A*). Self-adjointness thus means both symmetry and that the domain is maximal with respect to this relation, equivalently A is closed and A = A*, i.e., D(A) = D(A*).

Self-adjoint operators have important spectral and functional-analytic properties. They have real spectrum and admit a spectral theorem: there exists a projection-valued measure E such that A = ∫ λ dE(λ). This yields a functional calculus: for suitable functions f, f(A) is defined. They also generate strongly continuous one-parameter unitary groups via Stone’s theorem, with U(t) = e^{-itA}.

Essential notions include essential self-adjointness: a densely defined operator is essentially self-adjoint if its closure is self-adjoint, ensuring a unique self-adjoint extension. Examples arise in quantum mechanics, where observables are modeled by self-adjoint operators; differential operators like the Laplacian on a domain with appropriate boundary conditions illustrate how domain and boundary data influence self-adjointness. Von Neumann’s deficiency indices describe possible self-adjoint extensions of symmetric operators and underlie the classification of boundary conditions.