Essential spectrum in the Calkin algebra (${\sigma }_{\mathrm{ess}}(T)={\sigma }_{\mathcal{Q}(H)}([T])$)

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.