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.
Bounded self-adjoint operators on separable infinite-dimensional complex Hilbert spaces are unitarily equivalent modulo compact operators exactly when they have the same essential spectrum in the Calkin algebra. The Weyl-von Neumann theorem diagonalizes each modulo a compact error. Their diagonal cluster sets are the essential spectra; cluster-set matching of bounded sequences pairs the two diagonal lists with differences tending to zero, giving a compact diagonal difference. The separability assumption is essential: projections with countable and uncountable range dimensions can have the same essential spectrum but cannot differ by a compact operator after unitary conjugation.
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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.