# Calkin correspondence

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Calkin_correspondence
> Markdown URL: https://mediated.wiki/source/Calkin_correspondence.md
> Source: https://en.wikipedia.org/wiki/Calkin_correspondence
> Source revision: 1336393361
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

In mathematics, the **Calkin correspondence**, named after mathematician [John Williams Calkin](/source/John_Williams_Calkin), is a bijective correspondence between two-sided [ideals](/source/Ideal_(ring_theory)) of bounded [linear operators](/source/Linear_operators) of a separable infinite-dimensional [Hilbert space](/source/Hilbert_space) and Calkin sequence spaces (also called rearrangement invariant sequence spaces). The correspondence is implemented by mapping an operator to its [singular value](/source/Singular_value) sequence.

It originated from [John von Neumann](/source/John_von_Neumann)'s study of symmetric norms on [matrix algebras](/source/Linear_algebra).[1] It provides a fundamental classification and tool for the study of two-sided ideals of [compact operators](/source/Compact_operator) and their [traces](/source/Singular_trace), 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](/source/Sequence_space) *j* within *l*∞ can be embedded in *B*(*H*) using an arbitrary orthonormal basis {*e**n* }*n*=0∞. Associate to a sequence *a* from *j* the bounded operator

- - - - d i a g ( a ) = ∑ n = 0 ∞ a n | e n ⟩ ⟨ e n | , {\displaystyle {\rm {diag}}(a)=\sum _{n=0}^{\infty }a_{n}|e_{n}\rangle \langle e_{n}|,}

where [bra–ket notation](/source/Bra%E2%80%93ket_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](/source/Lorentz_space) of *a*. The decreasing rearrangement can be denoted μ(*n*,*a*), *n* = 0, 1, 2, ... Note that it is identical to the [singular values](/source/Singular_value) of the operator diag(*a*). Another notation for the decreasing rearrangement is *a**.

A *Calkin (or rearrangement invariant) sequence space* is a linear subspace *j* of the bounded sequences *l*∞ such that if *a* is a bounded sequence and μ(*n*,*a*) ≤ μ(*n*,*b*), *n* = 0, 1, 2, ..., for some *b* in *j*, then *a* belongs to *j*.

## Correspondence

Associate to a two-sided ideal *J* the sequence space *j* given by

- - - - j = { a ∈ l ∞ : d i a g ( μ ( a ) ) ∈ J } . {\displaystyle j=\{a\in l_{\infty }:{\rm {diag}}(\mu (a))\in J\}.}

Associate to a sequence space *j* the two-sided ideal *J* given by

- - - - J = { A ∈ B ( H ) : μ ( A ) ∈ j } . {\displaystyle J=\{A\in B(H):\mu (A)\in j\}.}

Here μ(*A*) and μ(*a*) are the [singular values](/source/Singular_value) of the operators *A* and diag(*a*), respectively. Calkin's Theorem[2] states that the two maps are inverse to each other. We obtain,

- **Calkin correspondence:** The two-sided ideals of [bounded operators](/source/Linear_operator) on an infinite dimensional separable Hilbert space and the Calkin sequence spaces are in bijective correspondence.

It is sufficient to know the association only between positive operators and positive sequences, hence the map μ: *J*+ → *j*+ from a positive operator to its [singular values](/source/Singular_value) implements the Calkin correspondence.

Another way of interpreting the Calkin correspondence, since the sequence space *j* is equivalent as a Banach space to the operators in the operator ideal *J* that are diagonal with respect to an arbitrary orthonormal basis, is that two-sided ideals are completely determined by their diagonal operators.

## Examples

Suppose *H* is a separable infinite-dimensional Hilbert space.

- **[Bounded operators](/source/Linear_operator).** The improper two-sided ideal *B*(*H*) corresponds to *l*∞.

- **[Compact operators](/source/Compact_operator).** The proper and norm closed two-sided ideal *K*(*H*) corresponds to *c*0, the [space of sequences converging to zero](/source/Sequence_space).

- **[Finite rank operators](/source/Finite-rank_operator).** The smallest two-sided ideal *F*(*H*) of finite rank operators corresponds to *c*00, the space of sequences with finite non-zero terms.

- **[Schatten *p*-ideals](/source/Schatten_class_operator).** The Schatten *p*-ideals *L**p*, *p* ≥ 1, correspond to the [*l**p* sequence spaces](/source/Sequence_space). In particular, the trace class operators correspond to *l**1* and the Hilbert-Schmidt operators correspond to *l**2* .

- **Weak-*L**p* ideals.** The weak-*L**p* ideals *L**p*,∞, *p* ≥ 1, correspond to the [weak-*l*p sequence spaces](/source/Lp_space).

- **Lorentz ψ-ideals.** The Lorentz ψ-ideals for an increasing concave function ψ : [0,∞) → [0,∞) correspond to the [Lorentz sequence spaces](/source/Lorentz_space).

## Notes

1. **[^](#cite_ref-vN1_1-0)** J. von Neumann (1937). "Some matrix inequalities and metrization of matrix space". *Tomsk. University Review*. **1**: 286–300.

1. **[^](#cite_ref-Ca1_2-0)** J. W. Calkin (1941). "Two-sided ideals and congruences in the ring of bounded operators in Hilbert space". *Ann. Math*. 2. **42** (4): 839–873. [doi](/source/Doi_(identifier)):[10.2307/1968771](https://doi.org/10.2307%2F1968771). [JSTOR](/source/JSTOR_(identifier)) [1968771](https://www.jstor.org/stable/1968771).

## References

- B. Simon (2005). *Trace ideals and their applications*. Providence, Rhode Island: Amer. Math. Soc. [ISBN](/source/ISBN_(identifier)) [978-0-8218-3581-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8218-3581-4).

- S. Lord, F. A. Sukochev. D. Zanin (2012). [*Singular traces: theory and applications*](http://www.degruyter.com/view/product/177778). Berlin: De Gruyter. [ISBN](/source/ISBN_(identifier)) [978-3-11-026255-1](https://en.wikipedia.org/wiki/Special:BookSources/978-3-11-026255-1).

---
Adapted from the Wikipedia article [Calkin correspondence](https://en.wikipedia.org/wiki/Calkin_correspondence) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Calkin_correspondence?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.