Self-adjointness refers to an important property of certain linear operators on a Hilbert space, where the operator is equal to its adjoint. This means that for an operator A, it satisfies the condition $$\langle Ax, y \rangle = \langle x, Ay \rangle$$ for all vectors x and y in the space. Self-adjoint operators play a crucial role in quantum mechanics and functional analysis, as they ensure real eigenvalues and orthogonal eigenvectors, which are essential for stability and predictability in physical systems.
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