Sselfadjointness

Selfadjointness is a property of linear operators acting on Hilbert spaces. A linear operator $A$ on a Hilbert space $H$ is called selfadjoint if its domain $D(A)$ is equal to its domain of definition $D(A^*)$, and for all vectors $u, v$ in $D(A)$, the following equality holds: $\langle Au, v \rangle = \langle u, Av \rangle$. Here, $A^*$ denotes the adjoint of the operator $A$, and $\langle \cdot, \cdot \rangle$ represents the inner product of the Hilbert space.

In simpler terms, a selfadjoint operator is one that is equal to its own adjoint. This property is crucial in quantum mechanics, where observables (like position, momentum, or energy) are represented by selfadjoint operators on a Hilbert space of quantum states. The spectral theorem for selfadjoint operators states that such operators have a complete set of orthogonal eigenvectors, whose corresponding eigenvalues are real numbers. These real eigenvalues represent the possible outcomes of measurements of the observable.

The condition that $D(A) = D(A^*)$ is important because the adjoint operator $A^*$ might be defined on a larger set than $A$. For an operator to be selfadjoint, its domain must precisely match the domain of its adjoint, and the equality of inner products must hold. Unbounded operators can also be selfadjoint, but their domains require careful consideration. If an operator is defined on a finite-dimensional inner product space, then selfadjointness is equivalent to the matrix representation of the operator being equal to its conjugate transpose.