Classification of separable self-adjoint operators modulo compacts

{title2=$\sigma_{\mathrm{ess}}(T)=\sigma_{\mathcal Q(H)}([T])$}

For a <bounded linear operator> on an infinite-dimensional complex <Hilbert space>, define the essential spectrum by failure of the <Fredholm> property for $T-\lambda I$. The <Atkinson theorem> identifies it with the spectrum in the <Calkin algebra>, hence it is compact and nonempty. On a finite-dimensional <Hilbert space> it is empty. For a bounded real diagonal operator on a separable infinite-dimensional space it equals the subsequential cluster set of the diagonal entries, including values of infinite multiplicity.


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 Incoming links: Essential spectrum in the Calkin algebra