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  "body": "\n## Statement\n\nLet $H$ be a [[separable|0Z2P41SVDP]] complex [[Hilbert space|HM2HHE3J51]], and let $\\varepsilon \u003e 0$.\nLet $A$ be a bounded operator on $H$ with $A = A^*$, where $A^*$ is the [[adjoint|6HSJBJ8TTQ]] of $A$.\nThen there are an [[orthonormal basis|0S9GGTR0EX]] $(e_n)$ of $H$, real numbers $(\\lambda_n)$ and a [[compact operator|M2NEP1K5AE]] $K$ on $H$ such that:\n\n1. $K = K^*$ and $\\norm{K} \u003c \\varepsilon$;\n2. $(A + K) e_n = \\lambda_n e_n$ for every $n$.\n\n## Remarks\n\nIn words, every bounded self-adjoint operator is the sum of a diagonal operator and a compact self-adjoint operator of arbitrarily small norm.\nWeyl proved the version with $K$ compact, and von Neumann showed that $K$ can be chosen in the Hilbert–Schmidt class, the [[Schatten class|153N5DW55Z]] $S_2$, with Hilbert–Schmidt norm less than $\\varepsilon$.\nThe idea of the proof is to use spectral projections of $A$ for small intervals.\nThey give finite-dimensional subspaces that are almost invariant under $A$, and $A$ is diagonalized up to a small error on each of them.\nExample: the multiplication operator $(Af)(x) = x f(x)$ on $L^2([0,1])$ has no eigenvalues at all, yet by the theorem a compact perturbation of small norm has an orthonormal basis of eigenvectors.\nThe theorem does not extend to trace class perturbations.\nBy the Kato–Rosenblum theorem, if $A$ has a nonzero absolutely continuous part, then so does $A + K$ for every trace class self-adjoint $K$.\nThen $A + K$ is not diagonal.\n\n## Explanations\n\n### L1 human\n\nA self-adjoint matrix can be diagonalized exactly, but a self-adjoint operator on an infinite-dimensional space may have no eigenvectors at all.\nThe theorem says that this failure is small: after adding a tiny compact correction, there is a full orthonormal basis of eigenvectors.\nSo, up to compact errors, every self-adjoint operator is diagonal.\n\n### L2 human\n\nThe theorem shows that the point spectrum and the continuous spectrum of a self-adjoint operator are not stable under compact perturbations, while the essential spectrum is, by Weyl's theorem.\nIt is a starting point of the theory of unitary equivalence modulo compact operators, which later led to the Brown–Douglas–Fillmore theory of extensions of C*-algebras.\n",
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