Let \(H\) be an infinite-dimensional complex Hilbert space, let \({\mathcal K}(H)\) be the ideal of compact operators, and let \({\mathcal C}(H)={\mathcal B}(H)/{\mathcal K}(H)\) be a Calkin algebra. Using recently obtained results by \textit{Y. F. Lin} and \textit{M. Mathieu} [Q. J. Math. 58, No. 2, 249--253 (2007; Zbl 1128.46021)], the authors show that any linear map on \({\mathcal B}(H)\), which is surjective up to compact operators and preserves essential spectral radius, induces a well-defined continuous bijection \(\phi: {\mathcal C}(H)\to{\mathcal C}(H)\). Moreover, \(\phi\) is an isomorphism or an antiisomorphism, possibly multiplied by a unimodular complex scalar. As an application, they classify linear maps, surjective up to compact operators, which preserve (i)~the essential spectrum, or (ii)~the left/right essential spectrum.