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In mathematics, the '''commutator subspace''' of a two-sided [[ideal]] of [[linear operators|bounded linear operators]] on a separable [[Hilbert space]] is the linear subspace spanned by [[commutator#Ring theory|commutators]] of operators in the ideal with bounded operators.
Modern characterisation of the commutator subspace is through the [[Calkin correspondence]] and it involves the invariance of the Calkin sequence space of an operator ideal to taking [[Cesàro mean]]s. This explicit spectral characterisation reduces problems and questions about commutators and [[singular trace|traces]] on two-sided ideals to (more resolvable) problems and conditions on sequence spaces.

== History ==

Commutators of linear operators on Hilbert spaces came to prominence in the 1930s as they featured in the [[matrix mechanics]], or Heisenberg, formulation of quantum mechanics. Commutator subspaces, though, received sparse attention until the 1970s. American mathematician [[Paul Halmos]] in 1954 showed that every bounded operator on a separable infinite dimensional Hilbert space is the sum of two commutators of bounded operators.<ref name="Ha2">{{cite journal
| author=P. Halmos
| year=1954
| url=http://www.jstor.org/discover/10.2307/2372409?uid=3737536&uid=2129&uid=2&uid=70&uid=4&sid=21102501655191
| title=Commutators of operators. II
| journal=Amer. J. Math.
| volume=76
| pages=191–198 }}
</ref>
In 1971 Carl Pearcy and David Topping revisited the topic and studied commutator subspaces for [[Schatten class operator|Schatten ideals]].<ref name="PT">
{{cite journal
| author=C. Pearcy and D. Topping
| url = http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.mmj/1029000686
| year=1971
| title=On commutators in ideals of compact operators
| journal=Michigan Math. J.
| volume=18
| pages=247–252 }}
</ref> As a student American mathematician Gary Weiss began to investigate spectral conditions for commutators of [[Hilbert–Schmidt operators]].<ref name="GW1">{{cite journal
| author= G. Weiss
| year=1980
| url=http://link.springer.com/article/10.1007%2FBF01702316#
| title=Commutators of Hilbert–Schmidt Operators, II
| journal=Integral Equations and Operator Theory
| volume=4
| pages=574–600 }}
</ref><ref name="GW2">{{cite journal
| author= G. Weiss
| year=1986
| url=http://link.springer.com/article/10.1007%2FBF01202521
| title=Commutators of Hilbert–Schmidt Operators, I
| journal=Integral Equations and Operator Theory
| volume=9
| pages=877–892 }}
</ref>
British mathematician [[Nigel Kalton]], noticing the spectral condition of Weiss, characterised all trace class commutators.<ref name="NK2">
{{cite journal
| author= N. J. Kalton
| year=1989
| url=http://kaltonmemorial.missouri.edu/docs/jfa1989.pdf
| title=Trace-class operators and commutators
| journal=J. Functional Analysis
| volume=86
| pages=41–74 }}</ref>
Kalton's result forms the basis for the modern characterisation of the commutator subspace.
In 2004 Ken Dykema, Tadeusz Figiel, Gary Weiss and Mariusz Wodzicki published the spectral characterisation of normal operators in the commutator subspace for every two-sided ideal of compact operators.<ref name="DFWW">
{{cite journal
| author= K. Dykema, T. Figiel, G. Weiss, M. Wodzicki
| url=http://math.berkeley.edu/~wodzicki/prace/Advances-185.pdf
| year=2004
| title=Commutator structure of operator ideals
| journal=Adv. Math.
| volume=185
| pages=1–79 }}
</ref>

== Definition ==

The commutator subspace of a two-sided ideal ''J'' of the bounded linear operators ''B''(''H'') on a separable Hilbert space ''H'' is the linear span of operators in ''J'' of the form [''A'',''B''] = ''AB'' − ''BA'' for all operators ''A'' from ''J'' and ''B'' from ''B''(''H'').

The commutator subspace of ''J'' is a linear subspace of ''J'' denoted by Com(''J'') or [''B''(''H''),''J''].

== Spectral characterisation ==

The [[Calkin correspondence]] states that a [[compact operator]] ''A'' belongs to a two-sided ideal ''J'' if and only if the [[singular values]] μ(''A'') of ''A'' belongs to the Calkin sequence space ''j'' associated to ''J''. [[Normal operator]]s that belong to the commutator subspace Com(''J'') can characterised as those ''A'' such that μ(''A'') belongs to ''j'' ''and'' the [[Cesàro mean]] of the sequence μ(''A'') belongs to ''j''.<ref name="DFWW"/> The following theorem is a slight extension to differences of normal operators<ref name="KLPS">
{{cite journal
|author=N. J. Kalton, S. Lord, D. Potapov, F. Sukochev
|title=Traces of compact operators and the noncommutative residue
|journal=Adv. Math.
|year=2013
|volume=235
|pages=1–55
|url=http://kaltonmemorial.missouri.edu/docs/adv2013.pdf }}
</ref> (setting ''B'' {{=}} 0 in the following gives the statement of the previous sentence).

: '''Theorem.''' Suppose ''A,B'' are compact normal operators that belong to a two-sided ideal ''J''. Then ''A'' − ''B'' belongs to the commutator subspace Com(''J'') if and only if
::::<math> \left\{ \frac{1}{1+n} \sum_{k=0}^n \left( \mu(k,A) - \mu(k,B) \right) \right\}_{n=0}^\infty \in j </math>
:where ''j'' is the Calkin sequence space corresponding to ''J'' and μ(''A''), μ(''B'') are the singular values of ''A'' and ''B'', respectively.

Provided that the [[eigenvalue|eigenvalue sequences]] of all operators in ''J'' belong to the Calkin sequence space ''j'' there is a spectral characterisation for arbitrary (non-normal) operators. It is not valid for every two-sided ideal but necessary and sufficient conditions are known. Nigel Kalton and American mathematician Ken Dykema introduced the condition first for countably generated ideals.<ref name="N2">
{{cite journal
|author=N. J. Kalton
|title=Spectral characterization of sums of commutators, I
|journal=J. Reine Angew. Math.
|year=1998
|volume=504
|pages=115–125 }}</ref><ref name="DK">
{{cite journal
|author=K. Dykema, N. J. Kalton
|title=Spectral characterization of sums of commutators, II
|journal=J. Reine Angew. Math.
|year=1998
|volume=504
|pages=127–137 }}
</ref>
Uzbek and Australian mathematicians Fedor Sukochev and Dmitriy Zanin completed the eigenvalue characterisation.<ref name="SZ1">{{citation needed|date=September 2013}}
</ref>

: '''Theorem.''' Suppose ''J'' is a two-sided ideal such that a bounded operator ''A'' belongs to ''J'' whenever there is a bounded operator ''B'' in ''J'' such that
{{NumBlk|::::| <math> \prod_{k=0}^n \mu(k,A) \leq \prod_{k=0}^n \mu(k,B), \quad n=0,1,2, \ldots . </math> | {{EquationRef|1}} }}
:If the bounded operator ''A'' and ''B'' belong to ''J'' then ''A'' − ''B'' belongs to the commutator subspace Com(''J'') if and only if
::::<math> \left\{ \frac{1}{1+n} \sum_{k=0}^n \left( \lambda(k,A) - \lambda(k,B) \right) \right\}_{n=0}^\infty \in j </math>
:where ''j'' is the Calkin sequence space corresponding to ''J'' and λ(''A''), λ(''B'') are the sequence of eigenvalues of the operators ''A'' and ''B'', respectively, rearranged so that the absolute value of the eigenvalues is decreasing.

Most two-sided ideals satisfy the condition in the Theorem, included all Banach ideals and quasi-Banach ideals.

== Consequences of the characterisation ==

* Every operator in ''J'' is a sum of commutators if and only if the corresponding Calkin sequence space ''j'' is invariant under taking [[Cesàro mean]]s. In symbols, Com(''J'') {{=}} ''J'' is equivalent to C(''j'') {{=}} ''j'', where C denotes the Cesàro operator on sequences.

* In any two-sided ideal the difference between a positive operator and its diagonalisation is a sum of commutators. That is, ''A'' − diag(μ(''A'')) belongs to Com(''J'') for every positive operator ''A'' in ''J'' where diag(μ(''A'')) is the diagonalisation of ''A'' in an arbitrary orthonormal basis of the separable Hilbert space ''H''.

* In any two-sided ideal satisfying ({{EquationNote|1}}) the difference between an arbitrary operator and its diagonalisation is a sum of commutators. That is, ''A'' − diag(λ(''A'')) belongs to Com(''J'') for every operator ''A'' in ''J'' where diag(λ(''A'')) is the diagonalisation of ''A'' in an arbitrary orthonormal basis of the separable Hilbert space ''H'' and λ(''A'') is an eigenvalue sequence.

* Every [[nilpotent operator|quasi-nilpotent operator]] in a two-sided ideal satisfying ({{EquationNote|1}}) is a sum of commutators.

== Application to traces ==

{{Main|Singular trace}}

A trace φ on a two-sided ideal ''J'' of ''B''(''H)'' is a linear functional φ:''J'' → ℂ that vanishes on Com(''J''). The consequences above imply

* The two-sided ideal ''J'' has a non-zero trace if and only if C(''j'') ≠ ''j''.

* φ(''A'') = φ∘diag(μ(''A'')) for every positive operator ''A'' in ''J'' where diag(μ(''A'')) is the diagonalisation of ''A'' in an arbitrary orthonormal basis of the separable Hilbert space ''H''. That is, traces on ''J'' are in direct correspondence with [[symmetric functional]]s on ''j''.

* In any two-sided ideal satisfying ({{EquationNote|1}}), φ(''A'') = φ∘diag(λ(''A'')) for every operator ''A'' in ''J'' where diag(λ(''A'')) is the diagonalisation of ''A'' in an arbitrary orthonormal basis of the separable Hilbert space ''H'' and λ(''A'') is an eigenvalue sequence.

* In any two-sided ideal satisfying ({{EquationNote|1}}), φ(''Q'')=0 for every [[nilpotent operator|quasi-nilpotent operator]] ''Q'' from ''J'' and every trace φ on ''J''.

== Examples ==

Suppose ''H'' is a separable infinite dimensional Hilbert space.

* '''Compact operators.''' The [[compact operator on hilbert space|compact linear operators]] ''K''(''H'') correspond to the space of converging to zero sequences, ''c''0. For a converging to zero sequence the [[Cesàro mean]]s converge to zero. Therefore C(''c''0) = ''c''0 and Com(''K''(''H'')) {{=}} ''K''(''H'').

* '''Finite rank operators.''' The [[finite-rank operator|finite rank operators]] ''F''(''H'') correspond to the space of sequences with finite non-zero terms, ''c''00. The condition
::::<math> \left\{ \frac{a_1 + a_2 + \cdots + a_n}{n} \right\}_{n=1}^\infty \in c_{00} </math>
:occurs if and only if
::::<math> a_1 + a_2 + \cdots + a_N = 0 </math>
:for the sequence (a1, a2, ... , aN, 0, 0 , ...) in ''c''00. The kernel of the [[trace class#Definition|operator trace]] Tr on ''F''(''H'') and the commutator subspace of the finite rank operators are equal, ker Tr {{=}} Com(''F''(''H'')) ⊊ ''F''(''H'').

* '''Trace class operators.''' The [[trace class operator]]s ''L''1 correspond to the [[sequence space|summable sequences]]. The condition
::::<math> \left\{ \frac{a_1 + a_2 + \cdots + a_n}{n} \right\}_{n=1}^\infty \in \ell_{1} </math>
:is stronger than the condition that a1 + a2 ... = 0. An example is the sequence with
::::<math> a_n = \frac{1}{n \log^2(n)} , \quad n \geq 2 . </math>
:and
::::<math> a_1 = - \sum_{n=2}^\infty a_n. </math>
which has sum zero but does not have a summable sequence of Cesàro means. Hence Com(''L''1) ⊊ ker Tr ⊊ ''L''1.

* '''Weak trace class operators'''. The [[weak trace-class operator|weak trace class operators]] ''L''1,∞ correspond to the [[Lp space|weak-''l''1 sequence space]]. From the condition
::::<math> \left\{ \frac{a_1 + a_2 + \cdots + a_n}{n} \right\}_{n=1}^\infty \in \ell_{1,\infty} </math>
:or equivalently
::::<math> \left\{ a_1 + a_2 + \cdots + a_n \right\}_{n=1}^\infty = O(1) </math>
it is immediate that Com(''L''''1'',∞)+ {{=}} (''L''1)+. The commutator subspace of the weak trace class operators contains the trace class operators. The [[harmonic series|harmonic sequence]]
1,1/2,1/3,...,1/n,... belongs to ''l''1,∞ and it is has a divergent series, and therefore the
Cesàro means of the harmonic sequence do not belong to ''l''1,∞.
In summary, ''L''1 ⊊ Com(''L''1,∞) ⊊ ''L''1,∞.

== Notes ==

{{reflist}}

== References ==

* {{cite journal
| author= K. Dykema, T. Figiel, G. Weiss, M. Wodzicki
| url=http://math.berkeley.edu/~wodzicki/prace/Advances-185.pdf
| year=2004
| title=Commutator structure of operator ideals
| journal=Adv. Math.
| volume=185
| pages=1–79 }}

* {{Citation
| author=G. Weiss
| contribution=''B''(''H'')-commutators: a historical survey
| editor = Dumitru Gaşpar, Dan Timotin, László Zsidó, Israel Gohberg, Florian-Horia Vasilescu
| title = Recent Advances in Operator Theory, Operator Algebras, and their Applications
| series= Operator Theory: Advances and Applications
| volume=153
| pages=307–320
| publisher = Birkhäuser Basel
| place = Berlin
| year = 2005
| isbn=978-3-7643-7127-2 }}

* {{Citation
| author=T. Figiel, N. Kalton
| contribution=Symmetric linear functionals on function spaces
| editor = M. Cwikel, M. Englis, A. Kufner, L.-E. Persson, G. Sparr,
| title = Function Spaces, Interpolation Theory, and Related Topics: Proceedings of the International Conference in Honour of Jaak Peetre on His 65th Birthday : Lund, Sweden, August 17–22, 2000
| series= De Gruyter: Proceedings in Mathematics
| pages=311–332
| publisher = De Gruyter
| place = Berlin
| year = 2002
| isbn=978-3-11-019805-8 }}

* {{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:Traces]]
[[Category:Hilbert space]]
[[Category:Von Neumann algebras]]

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