Commutator subspace

<h2 id="history">History</h2>
Commutators of linear operators on Hilbert spaces came to prominence in the 1930s as they featured in the <a href="/facts/Matrix_mechanics/nJxC4pKP">matrix mechanics</a>, or Heisenberg, formulation of quantum mechanics. Commutator subspaces, though, received sparse attention until the 1970s. American mathematician <a href="/facts/Paul_Halmos/mR11lcap">Paul Halmos</a> in 1954 showed that every bounded operator on a separable infinite dimensional Hilbert space is the sum of two commutators of bounded operators.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>
In 1971 Carl Pearcy and David Topping revisited the topic and studied commutator subspaces for <a href="/facts/Schatten_class_operator/nDX0guuz">Schatten ideals</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> As a student American mathematician Gary Weiss began to investigate spectral conditions for commutators of <a href="/facts/Hilbert%E2%80%93Schmidt_operators/Xhy17yn6">Hilbert–Schmidt operators</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a><a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>
British mathematician <a href="/facts/Nigel_Kalton/QG63W3jY">Nigel Kalton</a>, noticing the spectral condition of Weiss, characterised all trace class commutators.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>
Kalton's result forms the basis for the modern characterisation of the commutator subspace.
In 2004 Ken Dykema, <a href="/facts/Tadeusz_Figiel/vzhKDeBS">Tadeusz Figiel</a>, Gary Weiss and <a href="/facts/Mariusz_Wodzicki/4V0FqHX7">Mariusz Wodzicki</a> published the spectral characterisation of normal operators in the commutator subspace for every two-sided ideal of compact operators.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

<h2 id="definition">Definition</h2>
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].

<h2 id="spectral-characterisation">Spectral characterisation</h2>
The <a href="/facts/Calkin_correspondence/8Lhv2xDg">Calkin correspondence</a> states that a <a href="/facts/Compact_operator/KNkcK4mE">compact operator</a> A belongs to a two-sided ideal J if and only if the <a href="/facts/Singular_values/0QWHnqGG">singular values</a> μ(A) of A belongs to the Calkin sequence space j associated to J. <a href="/facts/Normal_operator/lwtz7tuF">Normal operators</a> that belong to the commutator subspace Com(J) can characterised as those A such that μ(A) belongs to j and the <a href="/facts/Ces%C3%A0ro_mean/ttbvyscp">Cesàro mean</a> of the sequence μ(A) belongs to j.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> The following theorem is a slight extension to differences of normal operators<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> (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

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    {\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 <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalue sequences</a> 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.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a><a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>
Uzbek and Australian mathematicians Fedor Sukochev and Dmitriy Zanin completed the eigenvalue characterisation.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>

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
<table><tbody><tr><td> ∏ 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 .} </td> <td></td> <td>1</td></tr></tbody></table>
If the bounded operator A and B belong to J then A − B belongs to the commutator subspace Com(J) if and only if

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    {\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 of the eigenvalues is decreasing.
Most two-sided ideals satisfy the condition in the Theorem, included all Banach ideals and quasi-Banach ideals.

<h2 id="consequences-of-the-characterisation">Consequences of the characterisation</h2>
<ul><li>Every operator in J is a sum of commutators if and only if the corresponding Calkin sequence space j is invariant under taking <a href="/facts/Ces%C3%A0ro_mean/ttbvyscp">Cesàro means</a>. In symbols, Com(J) = J is equivalent to C(j) = j, where C denotes the Cesàro operator on sequences.</li>
<li>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.</li>
<li>In any two-sided ideal satisfying (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.</li>
<li>Every <a href="/facts/Nilpotent_operator/SUuFrMhE">quasi-nilpotent operator</a> in a two-sided ideal satisfying (1) is a sum of commutators.</li></ul>
<h2 id="application-to-traces">Application to traces</h2>
Main article: <a href="/facts/Singular_trace/S4xkdjCI">Singular trace</a>
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

<ul><li>The two-sided ideal J has a non-zero trace if and only if C(j) ≠ j.</li>
<li>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 functionals on j.</li>
<li>In any two-sided ideal satisfying (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.</li>
<li>In any two-sided ideal satisfying (1), φ(Q) = 0 for every <a href="/facts/Nilpotent_operator/SUuFrMhE">quasi-nilpotent operator</a> Q from J and every trace φ on J.</li></ul>
<h2 id="examples">Examples</h2>
Suppose H is a separable infinite dimensional Hilbert space.

<ul><li>Compact operators. The <a href="/facts/Compact_operator_on_hilbert_space/xaL1NE6M">compact linear operators</a> K(H) correspond to the space of converging to zero sequences, c0. For a converging to zero sequence the <a href="/facts/Ces%C3%A0ro_mean/ttbvyscp">Cesàro means</a> converge to zero. Therefore, C(c0) = c0 and Com(K(H)) = K(H).</li>
<li>Finite rank operators. The <a href="/facts/Finite-rank_operator/jA2kCmCA">finite rank operators</a> F(H) correspond to the space of sequences with finite non-zero terms, c00. The condition</li></ul>

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    {\displaystyle \left\{{\frac {a_{1}+a_{2}+\cdots +a_{n}}{n}}\right\}_{n=1}^{\infty }\in c_{00}}

occurs if and only if

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    {\displaystyle a_{1}+a_{2}+\cdots +a_{N}=0}

for the sequence (a1, a2, ... , aN, 0, 0 , ...) in c00. The kernel of the <a href="/facts/Trace_class/2AmWENA3">operator trace</a> Tr on F(H) and the commutator subspace of the finite rank operators are equal, ker Tr = Com(F(H)) ⊊ F(H).
<ul><li>Trace class operators. The <a href="/facts/Trace_class_operator/2AmWENA3">trace class operators</a> L1 correspond to the <a href="/facts/Sequence_space/2HkcM12K">summable sequences</a>. The condition</li></ul>

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    {\displaystyle \left\{{\frac {a_{1}+a_{2}+\cdots +a_{n}}{n}}\right\}_{n=1}^{\infty }\in \ell _{1}}

is stronger than the condition that a1 + a2 ... = 0. An example is the sequence with

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    {\displaystyle a_{n}={\frac {1}{n\log ^{2}(n)}},\quad n\geq 2.}

and

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    {\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(L1) ⊊ ker Tr ⊊ L1.

<ul><li>Weak trace class operators. The <a href="/facts/Weak_trace-class_operator/G0Sgt7yu">weak trace class operators</a> L1,∞ correspond to the <a href="/facts/Lp_space/gbad0Rv1">weak-l1 sequence space</a>. From the condition</li></ul>

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    {\displaystyle \left\{{\frac {a_{1}+a_{2}+\cdots +a_{n}}{n}}\right\}_{n=1}^{\infty }\in \ell _{1,\infty }}

or equivalently

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    {\displaystyle \left\{a_{1}+a_{2}+\cdots +a_{n}\right\}_{n=1}^{\infty }=O(1)}

it is immediate that Com(L1,∞)+ = (L1)+. The commutator subspace of the weak trace class operators contains the trace class operators. The <a href="/facts/Harmonic_series_(mathematics)/8KGDHKTg">harmonic sequence</a>
1,1/2,1/3,...,1/n,... belongs to l1,∞ and it has a divergent series, and therefore the
Cesàro means of the harmonic sequence do not belong to l1,∞.
In summary, L1 ⊊ Com(L1,∞) ⊊ L1,∞.

<h2 id="notes">Notes</h2>

<ul><li>K. Dykema; T. Figiel; G. Weiss; M. Wodzicki (2004). <a href="http://math.berkeley.edu/~wodzicki/prace/Advances-185.pdf">"Commutator structure of operator ideals"</a> (PDF). <a href="/facts/Advances_in_Mathematics/kXtcl4pM">Advances in Mathematics</a>. 185: 1–79. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2Fs0001-8708%2803%2900141-5">10.1016/s0001-8708(03)00141-5</a>.</li></ul>
<ul><li>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, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-7643-7127-2</li></ul>
<ul><li>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, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-11-019805-8</li></ul>
<ul><li>S. Lord, F. A. Sukochev. D. Zanin (2012). <a href="http://www.degruyter.com/view/product/177778">Singular traces: theory and applications</a>. Berlin: De Gruyter. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1515%2F9783110262551">10.1515/9783110262551</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-11-026255-1.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">P. Halmos (1954). "Commutators of operators. II". American Journal of Mathematics. 76 (1): 191–198. doi:10.2307/2372409. JSTOR 2372409. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">C. Pearcy; D. Topping (1971). "On commutators in ideals of compact operators". Michigan Mathematical Journal. 18 (3): 247–252. doi:10.1307/mmj/1029000686. <a href="http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.mmj/1029000686" target="_blank">http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.mmj/1029000686</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">G. Weiss (1980). "Commutators of Hilbert–Schmidt Operators, II". Integral Equations and Operator Theory. 3 (4): 574–600. doi:10.1007/BF01702316. S2CID 189875793. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">G. Weiss (1986). "Commutators of Hilbert–Schmidt Operators, I". Integral Equations and Operator Theory. 9 (6): 877–892. doi:10.1007/bf01202521. S2CID 122936389. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">N. J. Kalton (1989). "Trace-class operators and commutators". Journal of Functional Analysis. 86: 41–74. doi:10.1016/0022-1236(89)90064-5. <a href="https://doi.org/10.1016%2F0022-1236%2889%2990064-5" target="_blank">https://doi.org/10.1016%2F0022-1236%2889%2990064-5</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">K. Dykema; T. Figiel; G. Weiss; M. Wodzicki (2004). "Commutator structure of operator ideals" (PDF). Advances in Mathematics. 185: 1–79. doi:10.1016/s0001-8708(03)00141-5. <a href="http://math.berkeley.edu/~wodzicki/prace/Advances-185.pdf" target="_blank">http://math.berkeley.edu/~wodzicki/prace/Advances-185.pdf</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">K. Dykema; T. Figiel; G. Weiss; M. Wodzicki (2004). "Commutator structure of operator ideals" (PDF). Advances in Mathematics. 185: 1–79. doi:10.1016/s0001-8708(03)00141-5. <a href="http://math.berkeley.edu/~wodzicki/prace/Advances-185.pdf" target="_blank">http://math.berkeley.edu/~wodzicki/prace/Advances-185.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">N. J. Kalton; S. Lord; D. Potapov; F. Sukochev (2013). "Traces of compact operators and the noncommutative residue". Advances in Mathematics. 235: 1–55. arXiv:1210.3423. doi:10.1016/j.aim.2012.11.007. <a href="https://doi.org/10.1016%2Fj.aim.2012.11.007" target="_blank">https://doi.org/10.1016%2Fj.aim.2012.11.007</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">N. J. Kalton (1998). "Spectral characterization of sums of commutators, I". J. Reine Angew. Math. 1998 (504): 115–125. arXiv:math/9709209. doi:10.1515/crll.1998.102. S2CID 119124949. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">K. Dykema; N. J. Kalton (1998). "Spectral characterization of sums of commutators, II". J. Reine Angew. Math. 504: 127–137. <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">[citation needed] <a href="/wiki/Wikipedia:Citation_needed" target="_blank">/wiki/Wikipedia:Citation_needed</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
</ol>

Commutator subspace open-in-new

Commutator subspace