# Commutator subspace

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In mathematics, the **commutator subspace** of a two-sided [ideal](/source/Ideal_(ring_theory)) of [bounded linear operators](/source/Linear_operators) on a separable [Hilbert space](/source/Hilbert_space) is the [linear subspace](/source/Linear_subspace) spanned by [commutators](/source/Commutator#Ring_theory) of operators in the ideal with bounded operators. Modern characterisation of the commutator subspace is through the [Calkin correspondence](/source/Calkin_correspondence) and it involves the invariance of the Calkin sequence space of an [operator ideal](/source/Operator_ideal) to taking [Cesàro means](/source/Ces%C3%A0ro_mean). This explicit spectral characterisation reduces problems and questions about commutators and [traces](/source/Singular_trace) 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](/source/Matrix_mechanics), or Heisenberg, formulation of quantum mechanics. Commutator subspaces, though, received sparse attention until the 1970s. American mathematician [Paul Halmos](/source/Paul_Halmos) in 1954 showed that every [bounded operator](/source/Bounded_operator) on a separable infinite dimensional Hilbert space is the sum of two commutators of bounded operators.[1] In 1971 [Carl Pearcy](/source/Carl_Pearcy) and David Topping revisited the topic and studied commutator subspaces for [Schatten ideals](/source/Schatten_class_operator).[2] As a student American mathematician Gary Weiss began to investigate spectral conditions for commutators of [Hilbert–Schmidt operators](/source/Hilbert%E2%80%93Schmidt_operators).[3][4] British mathematician [Nigel Kalton](/source/Nigel_Kalton), noticing the spectral condition of Weiss, characterised all trace class commutators.[5] Kalton's result forms the basis for the modern characterisation of the commutator subspace. In 2004 Ken Dykema, [Tadeusz Figiel](/source/Tadeusz_Figiel), Gary Weiss and [Mariusz Wodzicki](/source/Mariusz_Wodzicki) published the spectral characterisation of normal operators in the commutator subspace for every two-sided ideal of compact operators.[6]

## 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](/source/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](/source/Calkin_correspondence) states that a [compact operator](/source/Compact_operator) *A* belongs to a two-sided ideal *J* [if and only if](/source/If_and_only_if) the [singular values](/source/Singular_values) μ(*A*) of *A* belongs to the Calkin sequence space *j* associated to *J*. [Normal operators](/source/Normal_operator) that belong to the commutator subspace Com(*J*) can characterised as those *A* such that μ(*A*) belongs to *j* *and* the [Cesàro mean](/source/Ces%C3%A0ro_mean) of the sequence μ(*A*) belongs to *j*.[6] The following theorem is a slight extension to differences of normal operators[7] (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 - - - { 1 1 + n ∑ k = 0 n ( μ ( k , A ) − μ ( k , B ) ) } n = 0 ∞ ∈ j {\displaystyle \left\{{\frac {1}{1+n}}\sum _{k=0}^{n}\left(\mu (k,A)-\mu (k,B)\right)\right\}_{n=0}^{\infty }\in j}

- 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 sequences](/source/Eigenvalue) 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.[8][9] Uzbek and Australian mathematicians Fedor Sukochev and Dmitriy Zanin completed the eigenvalue characterisation.[10]

- **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

∏ k = 0 n μ ( k , A ) ≤ ∏ k = 0 n μ ( k , B ) , n = 0 , 1 , 2 , … . {\displaystyle \prod _{k=0}^{n}\mu (k,A)\leq \prod _{k=0}^{n}\mu (k,B),\quad n=0,1,2,\ldots .} 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 - - - { 1 1 + n ∑ k = 0 n ( λ ( k , A ) − λ ( k , B ) ) } n = 0 ∞ ∈ j {\displaystyle \left\{{\frac {1}{1+n}}\sum _{k=0}^{n}\left(\lambda (k,A)-\lambda (k,B)\right)\right\}_{n=0}^{\infty }\in j}

- 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](/source/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 means](/source/Ces%C3%A0ro_mean). 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 (**[1](#math_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 [quasi-nilpotent operator](/source/Nilpotent_operator) in a two-sided ideal satisfying (**[1](#math_1)**) is a sum of commutators.

## Application to traces

Main article: [Singular trace](/source/Singular_trace)

A trace φ on a two-sided ideal *J* of *B*(*H)* is a linear functional φ:*J* → C {\displaystyle \mathbb {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* ∘ {\displaystyle \circ } 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 functionals](https://en.wikipedia.org/w/index.php?title=Symmetric_functional&action=edit&redlink=1) on *j*.

- In any two-sided ideal satisfying (**[1](#math_1)**), *φ*(*A*) = *φ* ∘ {\displaystyle \circ } 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 (**[1](#math_1)**), *φ*(*Q*) = 0 for every [quasi-nilpotent operator](/source/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 linear operators](/source/Compact_operator_on_hilbert_space) *K*(*H*) correspond to the space of converging to zero sequences, *c*0. For a converging to zero sequence the [Cesàro means](/source/Ces%C3%A0ro_mean) converge to zero. Therefore, C(*c*0) = *c*0 and Com(*K*(*H*)) = *K*(*H*).

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

- - - - { a 1 + a 2 + ⋯ + a n n } n = 1 ∞ ∈ c 00 {\displaystyle \left\{{\frac {a_{1}+a_{2}+\cdots +a_{n}}{n}}\right\}_{n=1}^{\infty }\in c_{00}}

- occurs if and only if - - - a 1 + a 2 + ⋯ + a N = 0 {\displaystyle a_{1}+a_{2}+\cdots +a_{N}=0}

- for the sequence (*a*1, *a*2, ... , *a**N*, 0, 0 , ...) in *c*00. The kernel of the [operator trace](/source/Trace_class#Definition) 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 operators](/source/Trace_class_operator) *L*1 correspond to the [summable sequences](/source/Sequence_space). The condition

- - - - { a 1 + a 2 + ⋯ + a n n } n = 1 ∞ ∈ ℓ 1 {\displaystyle \left\{{\frac {a_{1}+a_{2}+\cdots +a_{n}}{n}}\right\}_{n=1}^{\infty }\in \ell _{1}}

- is stronger than the condition that *a*1 + *a*2 ... = 0. An example is the sequence with - - - a n = 1 n log 2 ⁡ ( n ) , n ≥ 2. {\displaystyle a_{n}={\frac {1}{n\log ^{2}(n)}},\quad n\geq 2.}

- and - - - a 1 = − ∑ n = 2 ∞ a n . {\displaystyle a_{1}=-\sum _{n=2}^{\infty }a_{n}.}

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 operators](/source/Weak_trace-class_operator) *L*1,∞ correspond to the [weak-*l*1 sequence space](/source/Lp_space). From the condition

- - - - { a 1 + a 2 + ⋯ + a n n } n = 1 ∞ ∈ ℓ 1 , ∞ {\displaystyle \left\{{\frac {a_{1}+a_{2}+\cdots +a_{n}}{n}}\right\}_{n=1}^{\infty }\in \ell _{1,\infty }}

- or equivalently - - - { a 1 + a 2 + ⋯ + a n } n = 1 ∞ = O ( 1 ) {\displaystyle \left\{a_{1}+a_{2}+\cdots +a_{n}\right\}_{n=1}^{\infty }=O(1)}

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 sequence](/source/Harmonic_series_(mathematics)) 1,1/2,1/3,...,1/*n*,... belongs to *l*1,∞ and it 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

1. **[^](#cite_ref-Ha2_1-0)** P. Halmos (1954). "Commutators of operators. II". *American Journal of Mathematics*. **76** (1): 191–198. [doi](/source/Doi_(identifier)):[10.2307/2372409](https://doi.org/10.2307%2F2372409). [JSTOR](/source/JSTOR_(identifier)) [2372409](https://www.jstor.org/stable/2372409).

1. **[^](#cite_ref-PT_2-0)** C. Pearcy; D. Topping (1971). ["On commutators in ideals of compact operators"](http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.mmj/1029000686). *Michigan Mathematical Journal*. **18** (3): 247–252. [doi](/source/Doi_(identifier)):[10.1307/mmj/1029000686](https://doi.org/10.1307%2Fmmj%2F1029000686).

1. **[^](#cite_ref-GW1_3-0)** G. Weiss (1980). "Commutators of Hilbert–Schmidt Operators, II". *Integral Equations and Operator Theory*. **3** (4): 574–600. [doi](/source/Doi_(identifier)):[10.1007/BF01702316](https://doi.org/10.1007%2FBF01702316). [S2CID](/source/S2CID_(identifier)) [189875793](https://api.semanticscholar.org/CorpusID:189875793).

1. **[^](#cite_ref-GW2_4-0)** G. Weiss (1986). "Commutators of Hilbert–Schmidt Operators, I". *Integral Equations and Operator Theory*. **9** (6): 877–892. [doi](/source/Doi_(identifier)):[10.1007/bf01202521](https://doi.org/10.1007%2Fbf01202521). [S2CID](/source/S2CID_(identifier)) [122936389](https://api.semanticscholar.org/CorpusID:122936389).

1. **[^](#cite_ref-NK2_5-0)** N. J. Kalton (1989). ["Trace-class operators and commutators"](https://doi.org/10.1016%2F0022-1236%2889%2990064-5). *Journal of Functional Analysis*. **86**: 41–74. [doi](/source/Doi_(identifier)):[10.1016/0022-1236(89)90064-5](https://doi.org/10.1016%2F0022-1236%2889%2990064-5).

1. ^ [***a***](#cite_ref-DFWW_6-0) [***b***](#cite_ref-DFWW_6-1) K. Dykema; T. Figiel; G. Weiss; M. Wodzicki (2004). ["Commutator structure of operator ideals"](http://math.berkeley.edu/~wodzicki/prace/Advances-185.pdf) (PDF). *[Advances in Mathematics](/source/Advances_in_Mathematics)*. **185**: 1–79. [doi](/source/Doi_(identifier)):[10.1016/s0001-8708(03)00141-5](https://doi.org/10.1016%2Fs0001-8708%2803%2900141-5).

1. **[^](#cite_ref-KLPS_7-0)** N. J. Kalton; S. Lord; D. Potapov; F. Sukochev (2013). ["Traces of compact operators and the noncommutative residue"](https://doi.org/10.1016%2Fj.aim.2012.11.007). *[Advances in Mathematics](/source/Advances_in_Mathematics)*. **235**: 1–55. [arXiv](/source/ArXiv_(identifier)):[1210.3423](https://arxiv.org/abs/1210.3423). [doi](/source/Doi_(identifier)):[10.1016/j.aim.2012.11.007](https://doi.org/10.1016%2Fj.aim.2012.11.007).

1. **[^](#cite_ref-N2_8-0)** N. J. Kalton (1998). "Spectral characterization of sums of commutators, I". *J. Reine Angew. Math*. **1998** (504): 115–125. [arXiv](/source/ArXiv_(identifier)):[math/9709209](https://arxiv.org/abs/math/9709209). [doi](/source/Doi_(identifier)):[10.1515/crll.1998.102](https://doi.org/10.1515%2Fcrll.1998.102). [S2CID](/source/S2CID_(identifier)) [119124949](https://api.semanticscholar.org/CorpusID:119124949).

1. **[^](#cite_ref-DK_9-0)** K. Dykema; N. J. Kalton (1998). "Spectral characterization of sums of commutators, II". *J. Reine Angew. Math*. **504**: 127–137.

1. **[^](#cite_ref-SZ1_10-0)** [*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed)*]

## References

- K. Dykema; T. Figiel; G. Weiss; M. Wodzicki (2004). ["Commutator structure of operator ideals"](http://math.berkeley.edu/~wodzicki/prace/Advances-185.pdf) (PDF). *[Advances in Mathematics](/source/Advances_in_Mathematics)*. **185**: 1–79. [doi](/source/Doi_(identifier)):[10.1016/s0001-8708(03)00141-5](https://doi.org/10.1016%2Fs0001-8708%2803%2900141-5).

- G. Weiss (2005), "*B*(*H*)-commutators: a historical survey", in Dumitru Gaşpar; Dan Timotin; László Zsidó; Israel Gohberg; Florian-Horia Vasilescu (eds.), *Recent Advances in Operator Theory, Operator Algebras, and their Applications*, Operator Theory: Advances and Applications, vol. 153, Berlin: Birkhäuser Basel, pp. 307–320, [ISBN](/source/ISBN_(identifier)) [978-3-7643-7127-2](https://en.wikipedia.org/wiki/Special:BookSources/978-3-7643-7127-2)

- T. Figiel; N. Kalton (2002), "Symmetric linear functionals on function spaces", in M. Cwikel; M. Englis; A. Kufner; L.-E. Persson; G. Sparr (eds.), *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*, De Gruyter: Proceedings in Mathematics, Berlin: De Gruyter, pp. 311–332, [ISBN](/source/ISBN_(identifier)) [978-3-11-019805-8](https://en.wikipedia.org/wiki/Special:BookSources/978-3-11-019805-8)

- S. Lord, F. A. Sukochev. D. Zanin (2012). [*Singular traces: theory and applications*](http://www.degruyter.com/view/product/177778). Berlin: De Gruyter. [doi](/source/Doi_(identifier)):[10.1515/9783110262551](https://doi.org/10.1515%2F9783110262551). [ISBN](/source/ISBN_(identifier)) [978-3-11-026255-1](https://en.wikipedia.org/wiki/Special:BookSources/978-3-11-026255-1).

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