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 (also called rearrangement invariant 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 (more resolvable) problems on sequence spaces.

Definitions

A two-sided ideal J of the bounded linear operators B(H) on a separable Hilbert space H is a linear subspace such that AB and BA belong to J for all operators A from J and B from B(H).

A sequence space j within l∞ can be embedded in B(H) using an arbitrary orthonormal basis {en }n=0∞. Associate to a sequence a from j the bounded operator

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{\displaystyle {\rm {diag}}(a)=\sum _{n=0}^{\infty }a_{n}|e_{n}\rangle \langle e_{n}|,}

where bra–ket notation has been used for the one-dimensional projections onto the subspaces spanned by individual basis vectors. The sequence of absolute values of the entries of a in decreasing order is called the decreasing rearrangement of a. The decreasing rearrangement can be denoted μ(n,a), n = 0, 1, 2, ... Note that it is identical to the singular values of the operator diag(a).