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Complexity index - Revision history
2026-08-23T09:42:52Z
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en>Monkbot: /* References */Task 4: Fix CS1 deprecated coauthor parameter errors
2014-05-12T16:55:09Z
<p><span class="autocomment">References: </span>Task 4: Fix <a href="/w/index.php?title=Help:CS1_errors&action=edit&redlink=1" class="new" title="Help:CS1 errors (page does not exist)">CS1 deprecated coauthor parameter errors</a></p>
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en>Discospinster: Reverted edits by 69.146.143.26 (talk) to last version by Addbot
2013-10-24T01:14:44Z
<p><a href="/w/index.php?title=Help:Reverting&action=edit&redlink=1" class="new" title="Help:Reverting (page does not exist)">Reverted</a> edits by <a href="/wiki/Special:Contributions/69.146.143.26" title="Special:Contributions/69.146.143.26">69.146.143.26</a> (<a href="/w/index.php?title=User_talk:69.146.143.26&action=edit&redlink=1" class="new" title="User talk:69.146.143.26 (page does not exist)">talk</a>) to last version by Addbot</p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 03:14, 24 October 2013</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Hi! </del><<del style="font-weight: bold; text-decoration: none;">br</del>><del style="font-weight: bold; text-decoration: none;">My name </del>is <del style="font-weight: bold; text-decoration: none;">Lizette and I</del>'<del style="font-weight: bold; text-decoration: none;">m a 26 years old girl from Australia</del>.<<del style="font-weight: bold; text-decoration: none;">br</del>><del style="font-weight: bold; text-decoration: none;">xunjie つぼみのスカート鵬、</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">In mathematics, the '''Dixmier trace''', introduced by {{harvs |txt |authorlink=Jacques Dixmier |first=Jacques |last=Dixmier |year=1966}}, is a non-normal trace on a space of [[linear operator]]s on a [[Hilbert space]] larger than the space of [[trace class operator]]s. Dixmier traces are examples of [[singular trace]]s.</ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">ダイナミックなダンスをジャンプして色鮮やかなハイウエストのズボンを満たしている必要があっても、</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">受注から2週間以内に開始するように、 [http</del>://<del style="font-weight: bold; text-decoration: none;">www</del>.<del style="font-weight: bold; text-decoration: none;">almondpress.org</del>/<del style="font-weight: bold; text-decoration: none;">hot</del>/<del style="font-weight: bold; text-decoration: none;">fashion</del>/<del style="font-weight: bold; text-decoration: none;">chanel</del>.<del style="font-weight: bold; text-decoration: none;">html ����ͥ� ؔ�� ��ʽ] 彼らはこのシリーズの人形モデルの美し緩いウエストデザインを表現することを好むだろうと、</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Some applications of Dixmier traces to [[noncommutative geometry]] are described in {{harv|Connes|1994}}.</ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">バッグのようにカードや書類、</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">国の研究開発拠点を設置する最初のボディデザインにより、 </del>[<del style="font-weight: bold; text-decoration: none;">http:</del>//<del style="font-weight: bold; text-decoration: none;">www</del>.<del style="font-weight: bold; text-decoration: none;">schochauer</del>.<del style="font-weight: bold; text-decoration: none;">ch</del>/<del style="font-weight: bold; text-decoration: none;">_js</del>/<del style="font-weight: bold; text-decoration: none;">r</del>/<del style="font-weight: bold; text-decoration: none;">e</del>/<del style="font-weight: bold; text-decoration: none;">mall</del>/<del style="font-weight: bold; text-decoration: none;">watch</del>/<del style="font-weight: bold; text-decoration: none;">gaga</del>/ <del style="font-weight: bold; text-decoration: none;">�����ߥ�� �rӋ ���] あなたの健康日：2013年8月26日午後01時30分41秒この提案が実現可能であり、</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Definition==</ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">MQDマジデイトンデイトンの専門の男性の子供服ブランド、</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">If ''H'' is a Hilbert space, then ''L''</ins><<ins style="font-weight: bold; text-decoration: none;">sup</ins>><ins style="font-weight: bold; text-decoration: none;">1,∞</sup>(''H'') </ins>is <ins style="font-weight: bold; text-decoration: none;">the space of compact linear operators ''T'' on ''H'' such that the norm</ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">地質時代の燕の期間に、[http:</del>//<del style="font-weight: bold; text-decoration: none;">www</del>.<del style="font-weight: bold; text-decoration: none;">tobler</del>-<del style="font-weight: bold; text-decoration: none;">verlag</del>.<del style="font-weight: bold; text-decoration: none;">ch</del>/<del style="font-weight: bold; text-decoration: none;">flash</del>/<del style="font-weight: bold; text-decoration: none;">ja</del>/<del style="font-weight: bold; text-decoration: none;">li</del>/<del style="font-weight: bold; text-decoration: none;">top/gaga/ gaga milano �rӋ</del>] <del style="font-weight: bold; text-decoration: none;">ショッピングモールの新店舗で、</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">16000平方フィートを占めているというようにスーパーモデルリン、</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">:<math>\|T\|_{1,\infty} = \sup_N\frac{\sum_{i=1}^N \mu_i(T)}{\log(N)}</math></ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">全国の互いに結合投資する招待されています！ ①ベンの原稿がより多くの情報の普及のため、</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">保証の拡大に企業金融がブロックすることは困難であることを記者団に語った。 [</del>http://<del style="font-weight: bold; text-decoration: none;">athenavineyards</del>.<del style="font-weight: bold; text-decoration: none;">com</del>/<del style="font-weight: bold; text-decoration: none;">media</del>/<del style="font-weight: bold; text-decoration: none;">duvetica</del>.<del style="font-weight: bold; text-decoration: none;">php ���`�� �����ȥ�å�</del>]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">is finite, where the numbers ''&mu;''<sub>''i''</sub>(''T'') are the eigenvalues of |''T'</ins>'<ins style="font-weight: bold; text-decoration: none;">| arranged in decreasing order</ins>. <ins style="font-weight: bold; text-decoration: none;"> Let </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">:<math>a_N = \frac{\sum_{i=1}^N \mu_i(T)}{\log(N)}</ins><<ins style="font-weight: bold; text-decoration: none;">/math</ins>><ins style="font-weight: bold; text-decoration: none;">.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The Dixmier trace Tr<sub>''&omega;''</sub>(''T'') of ''T'' is defined for positive operators ''T'' of ''L''<sup>1,∞</sup>(''H'') to be </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:<ins style="font-weight: bold; text-decoration: none;"><math>\operatorname{Tr}_\omega(T)= \lim_\omega a_N<</ins>/<ins style="font-weight: bold; text-decoration: none;">math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">where lim<sub>''&omega;''<</ins>/<ins style="font-weight: bold; text-decoration: none;">sub> is a scale-invariant positive "extension" of the usual limit, to all bounded sequences</ins>. <ins style="font-weight: bold; text-decoration: none;"> In other words, it has the following properties:</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*lim<sub>''&omega;''</sub>(''&alpha;''<sub>''n''</sub>) ≥ 0 if all ''&alpha;''<sub>''n''</sub> ≥ 0 (positivity)</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*lim<sub>''&omega;''</sub>(''&alpha;''<sub>''n''</sub>) = lim(''&alpha;''<sub>''n''</sub>) whenever the ordinary limit exists</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*lim<sub>''&omega;''</sub>(''&alpha;''<sub>1</sub>, ''&alpha;''<sub>1<</ins>/<ins style="font-weight: bold; text-decoration: none;">sub>, ''&alpha;''<sub>2<</ins>/<ins style="font-weight: bold; text-decoration: none;">sub>, ''&alpha;''<sub>2<</ins>/<ins style="font-weight: bold; text-decoration: none;">sub>, ''&alpha;''<sub>3</sub>, </ins>.<ins style="font-weight: bold; text-decoration: none;">..) = lim<sub>ω</sub>(''&alpha;''<sub>''n''</sub>) (scale invariance)</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">There are many such extensions (such as a </ins>[<ins style="font-weight: bold; text-decoration: none;">[Banach limit]] of ''&alpha;''<sub>1</sub>, ''&alpha;''<sub>2</sub>, ''&alpha;''<sub>4<</ins>/<ins style="font-weight: bold; text-decoration: none;">sub>, ''&alpha;''<sub>8<</ins>/<ins style="font-weight: bold; text-decoration: none;">sub>,</ins>..<ins style="font-weight: bold; text-decoration: none;">.) so there are many different Dixmier traces. </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">As the Dixmier trace is linear, it extends by linearity to all operators of ''L''<sup>1,∞<</ins>/<ins style="font-weight: bold; text-decoration: none;">sup>(''H'').</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">If the Dixmier trace of an operator is independent of the choice of lim<sub>''&omega;''<</ins>/<ins style="font-weight: bold; text-decoration: none;">sub> then the operator is called '''measurable'''.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Properties==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*Tr<sub>''&omega;''<</ins>/<ins style="font-weight: bold; text-decoration: none;">sub>(''T'') is linear in ''T''.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*If ''T'' ≥ 0 then Tr<sub>''&omega;''<</ins>/<ins style="font-weight: bold; text-decoration: none;">sub>(''T'') ≥ 0</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*If ''S'' is bounded then Tr<sub>ω<</ins>/<ins style="font-weight: bold; text-decoration: none;">sub>(''ST'') = Tr<sub>''&omega;''<</ins>/<ins style="font-weight: bold; text-decoration: none;">sub>(''TS'')</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*Tr<sub>ω<</ins>/<ins style="font-weight: bold; text-decoration: none;">sub>(''T'') does not depend on the choice of inner product on ''H''.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*Tr<sub>''&omega;''</sub>(''T'') = 0 for all trace class operators ''T'', but there are compact operators for which it is equal to 1.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">A trace ''&phi;'' is called '''normal''' if ''&phi;''(sup ''x''<sub>α<</ins>/<ins style="font-weight: bold; text-decoration: none;">sub>) = sup&nbsp;''&phi;''( ''x''<sub>''&alpha;''<</ins>/<ins style="font-weight: bold; text-decoration: none;">sub>) for every bounded increasing directed family of positive operators</ins>. <ins style="font-weight: bold; text-decoration: none;"> Any normal trace on <math>L^{1,\infty}(H)</math> is equal to the usual trace, so the Dixmier trace is an example of a non</ins>-<ins style="font-weight: bold; text-decoration: none;">normal trace</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Examples==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">A compact self-adjoint operator with eigenvalues 1, 1</ins>/<ins style="font-weight: bold; text-decoration: none;">2, 1</ins>/<ins style="font-weight: bold; text-decoration: none;">3, ... has Dixmier trace equal to 1.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">If the eigenvalues μ<sub>i<</ins>/<ins style="font-weight: bold; text-decoration: none;">sub> of the positive operator ''T'' have the property that </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">:<math>\zeta_T(s)= \operatorname{Tr}(T^s)= \sum{\mu_i^s}<</ins>/<ins style="font-weight: bold; text-decoration: none;">math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">converges for Re(''s'')>1 and extends to a meromorphic function near ''s''=1 with at most a simple pole at ''s''=1, then the Dixmier trace</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">of ''T'' is the residue at ''s''=1 (and in particular is independent of the choice of ω).</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{harvtxt|Connes|1988}} showed that Wodzicki's [[noncommutative residue]] {{harv|Wodzicki|1984}} of a [[pseudodifferential operator]] on a [[manifold]</ins>] <ins style="font-weight: bold; text-decoration: none;">is equal to its Dixmier trace.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==References==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*Albeverio, S.; Guido, D.; Ponosov, A.; Scarlatti, S.: Singular traces and compact operators. J. Funct. Anal. 137 (1996), no.&nbsp;2, 281—302.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*{{Citation | doi=10.1007/BF01218391 | last1=Connes | first1=Alain | author1-link=Alain Connes | title=The action functional in noncommutative geometry | url=</ins>http://<ins style="font-weight: bold; text-decoration: none;">projecteuclid</ins>.<ins style="font-weight: bold; text-decoration: none;">org</ins>/<ins style="font-weight: bold; text-decoration: none;">euclid.cmp</ins>/<ins style="font-weight: bold; text-decoration: none;">1104161823 | mr=953826 | year=1988 | journal=Communications in Mathematical Physics | issn=0010-3616 | volume=117 | issue=4 | pages=673–683}}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*{{Citation | last1=Connes | first1=Alain | author1-link=Alain Connes | title=Non-commutative geometry | url=ftp://ftp</ins>.<ins style="font-weight: bold; text-decoration: none;">alainconnes.org/book94bigpdf.pdf | publisher=[[Academic Press]] | location=Boston, MA | isbn=978-0-12-185860-5 | year=1994}}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*{{Citation | last1=Dixmier | first1=Jacques | title=Existence de traces non normales | mr=0196508 | year=1966 | journal=Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Séries a et B | issn=0151-0509 | volume=262 | pages=A1107&ndash;A1108}}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*{{Citation | last1=Wodzicki | first1=M. | title=Local invariants of spectral asymmetry | doi=10.1007/BF01403095 | mr=728144 | year=1984 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=75 | issue=1 | pages=143–177}}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== See also ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [[Singular trace]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Von Neumann algebras]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Hilbert space]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Operator theory]</ins>]</div></td></tr>
</table>
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