What is Calkin correspondence?

In mathematics, the Calkin correspondence, named after mathematician John Williams Calkin, is a bijective correspondence between two-sided ideals of bounded linear operators of a separable infinite-dimensional Hilbert space and Calkin sequence spaces. The correspondence is implemented by mapping an operator to its singular value sequence.

It originated from John von Neumann's study of symmetric norms on matrix algebras. It provides a fundamental classification and tool for the study of two-sided ideals of compact operators and their traces, by reducing problems about operator spaces to problems on sequence spaces.