= Classification of separable self-adjoint operators modulo compacts
{title2=$T\sim S\iff\sigma_{\mathrm{ess}}(T)=\sigma_{\mathrm{ess}}(S)$}
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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