In mathematics, the '''Calkin correspondence''', named after mathematician [[John Williams Calkin]], is a bijective correspondence between two-sided [[ideal (ring theory)|ideal]]s 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 [[linear algebra|matrix algebras]].<ref name="vN1">{{cite journal | author= J. von Neumann | year=1937 | title=Some matrix inequalities and metrization of matrix space | volume = 1 | journal = Tomsk. University Review | pages= 286–300}}</ref> It provides a fundamental classification and tool for the study of two-sided ideals of [[compact operator]]s and their [[singular trace|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''<sub>∞</sub> can be embedded in ''B''(''H'') using an arbitrary orthonormal basis {''e''<sub>''n''</sub> }<sub>''n''=0</sub><sup>∞</sup>. Associate to a sequence ''a'' from ''j'' the bounded operator ::::<math> {\rm diag}(a) = \sum_{n=0}^\infty a_n | e_n \rangle \langle e_n |, </math> 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 [[Lorentz space|decreasing rearrangement]] of&nbsp;''a''. The decreasing rearrangement can be denoted μ(''n'',''a''), ''n''&nbsp;=&nbsp;0,&nbsp;1,&nbsp;2,&nbsp;... Note that it is identical to the [[singular value]]s of the operator diag(''a''). Another notation for the decreasing rearrangement is&nbsp;''a''*.

A ''Calkin (or rearrangement invariant) sequence space'' is a linear subspace ''j'' of the bounded sequences ''l''<sub>∞</sub> such that if ''a'' is a bounded sequence and μ(''n'',''a'') ≤ μ(''n'',''b''), ''n''&nbsp;{{=}}&nbsp;0,&nbsp;1,&nbsp;2,&nbsp;..., for some ''b'' in ''j'', then ''a'' belongs to&nbsp;''j''.

== Correspondence ==

Associate to a two-sided ideal ''J'' the sequence space ''j'' given by ::::<math> j = \{ a \in l_\infty : {\rm diag}(\mu(a)) \in J \} . </math> Associate to a sequence space ''j'' the two-sided ideal ''J'' given by ::::<math> J = \{ A \in B(H) : \mu(A) \in j \} . </math> Here μ(''A'') and μ(''a'') are the [[singular value]]s of the operators ''A'' and diag(''a''), respectively. Calkin's Theorem<ref name="Ca1">{{cite journal | author= J. W. Calkin | year=1941 | title=Two-sided ideals and congruences in the ring of bounded operators in Hilbert space | volume =42 | journal = Ann. Math. |series=2 | pages= 839–873 | doi= 10.2307/1968771 | issue= 4 | jstor=1968771 }}</ref> states that the two maps are inverse to each other. We obtain,

:'''Calkin correspondence:''' The two-sided ideals of [[linear operator|bounded operators]] 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''<sub>+</sub>&nbsp;→&nbsp;''j''<sub>+</sub> from a positive operator to its [[singular value]]s 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.

*'''[[linear operator|Bounded operators]].''' The improper two-sided ideal ''B''(''H'') corresponds to ''l''<sub>∞</sub>. *'''[[Compact operator]]s.''' The proper and norm closed two-sided ideal ''K''(''H'') corresponds to ''c''<sub>0</sub>, the [[sequence space|space of sequences converging to zero]]. *'''[[Finite-rank operator|Finite rank operators]].''' The smallest two-sided ideal ''F''(''H'') of finite rank operators corresponds to ''c''<sub>00</sub>, the space of sequences with finite non-zero terms. *'''[[Schatten class operator|Schatten ''p''-ideals]].''' The Schatten ''p''-ideals ''L''<sub>''p''</sub>, ''p''&nbsp;≥&nbsp;1, correspond to the [[sequence space|''l''<sub>''p''</sub> sequence spaces]]. In particular, the trace class operators correspond to ''l''<sub>''1''</sub> and the Hilbert-Schmidt operators correspond to ''l''<sub>''2''</sub> . * '''Weak-''L''<sub>''p''</sub> ideals.''' The weak-''L''<sub>''p''</sub> ideals ''L''<sub>''p'',∞</sub>, ''p''&nbsp;≥&nbsp;1, correspond to the [[Lp space|weak-''l''<sub>p</sub> sequence spaces]]. * '''Lorentz ψ-ideals.''' The Lorentz ψ-ideals for an increasing concave function ψ&nbsp;:&nbsp;[0,∞)&nbsp;→&nbsp;[0,∞) correspond to the [[Lorentz space|Lorentz sequence spaces]].

== Notes ==

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== References ==

* {{cite book | isbn=978-0-8218-3581-4 | author= B. Simon | year=2005 | title=Trace ideals and their applications | publisher=Amer. Math. Soc. | location=Providence, Rhode Island }}

* {{cite book | isbn=978-3-11-026255-1 | author= S. Lord, F. A. Sukochev. D. Zanin | year=2012 | url=http://www.degruyter.com/view/product/177778 | title=Singular traces: theory and applications | publisher=De Gruyter | location=Berlin }}

[[Category:Operator algebras]] [[Category:Hilbert spaces]] [[Category:Von Neumann algebras]]