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 (also called rearrangement invariant sequence spaces).
It originated from John von Neumann's study of symmetric norms on matrix algebras.
[1] 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 (more resolvable) problems on sequence spaces.
Associate to a sequence a from j the bounded operator where bra–ket notation has been used for the one-dimensional projections onto the subspaces spanned by individual basis vectors.
Associate to a two-sided ideal J the sequence space j given by Associate to a sequence space j the two-sided ideal J given by Here μ(A) and μ(a) are the singular values of the operators A and diag(a), respectively.