The Weyl–von Neumann theorem is a result in the theory of linear operators, particularly in the realm of functional analysis and operator theory. It addresses the spectral properties of self-adjoint or symmetric operators in Hilbert spaces. Specifically, the theorem characterizes the absolutely continuous spectrum of a bounded self-adjoint operator.
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Every self-adjoint operator on a separable complex Hilbert space is diagonal modulo a compact operator. For a bounded operator the compact self-adjoint error can have arbitrarily small Hilbert-Schmidt norm. In a cyclic spectral representation, partition a bounded real interval into intervals of width at most and successively bisect. Orthogonal step-function differences give basis vectors supported in the parent intervals. A diagonal value chosen in each support interval gives total squared column error at most . Since , this tends to zero. Apply the construction to countably many cyclic summands with summable squared error bounds. For an unbounded operator first split into bounded spectral intervals, obtaining a bounded compact error on the same domain.