A symmetric operator is a type of linear operator defined on a domain in a Hilbert space that satisfies the condition \( \langle Ax, y \rangle = \langle x, Ay \rangle \) for all elements \( x \) and \( y \) in its domain. This property ensures that the operator is self-adjoint when it is equal to its adjoint and plays a critical role in the study of unbounded linear operators and their spectral properties.
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