466:
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. Uzbek and
816:
408:
2095:
1735:
64:
in 1954 showed that every bounded operator on a separable infinite dimensional
Hilbert space is the sum of two commutators of bounded operators. In 1971 Carl Pearcy and David Topping revisited the topic and studied commutator subspaces for
1451:
630:
2211:
1840:
77:, noticing the spectral condition of Weiss, characterised all trace class commutators. Kalton's result forms the basis for the modern characterisation of the commutator subspace. In 2004 Ken Dykema,
1921:
682:
2702:
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
1530:
274:
1987:
1063:
1633:
1349:
503:
1136:
2700:
T. Figiel; N. Kalton (2002), "Symmetric linear functionals on function spaces", in M. Cwikel; M. Englis; A. Kufner; L.-E. Persson; G. Sparr (eds.),
2343:
60:, or Heisenberg, formulation of quantum mechanics. Commutator subspaces, though, received sparse attention until the 1970s. American mathematician
2114:
85:
published the spectral characterisation of normal operators in the commutator subspace for every two-sided ideal of compact operators.
1768:
2680:)-commutators: a historical survey", in Dumitru Gaşpar; Dan Timotin; László Zsidó; Israel Gohberg; Florian-Horia Vasilescu (eds.),
2739:
2709:
2689:
902:
In any two-sided ideal the difference between a positive operator and its diagonalisation is a sum of commutators. That is,
811:{\displaystyle \left\{{\frac {1}{1+n}}\sum _{k=0}^{n}\left(\lambda (k,A)-\lambda (k,B)\right)\right\}_{n=0}^{\infty }\in j}
1859:
1261:
36:
of operators in the ideal with bounded operators. Modern characterisation of the commutator subspace is through the
403:{\displaystyle \left\{{\frac {1}{1+n}}\sum _{k=0}^{n}\left(\mu (k,A)-\mu (k,B)\right)\right\}_{n=0}^{\infty }\in j}
1470:
862:
Most two-sided ideals satisfy the condition in the
Theorem, included all Banach ideals and quasi-Banach ideals.
2763:
2248:
70:
69:. As a student American mathematician Gary Weiss began to investigate spectral conditions for commutators of
2090:{\displaystyle \left\{{\frac {a_{1}+a_{2}+\cdots +a_{n}}{n}}\right\}_{n=1}^{\infty }\in \ell _{1,\infty }}
56:
Commutators of linear operators on
Hilbert spaces came to prominence in the 1930s as they featured in the
2684:, Operator Theory: Advances and Applications, vol. 153, Berlin: Birkhäuser Basel, pp. 307–320,
951:) the difference between an arbitrary operator and its diagonalisation is a sum of commutators. That is,
2262:
and it has a divergent series, and therefore the Cesàro means of the harmonic sequence do not belong to
467:
Australian mathematicians Fedor
Sukochev and Dmitriy Zanin completed the eigenvalue characterisation.
1954:
2247:. The commutator subspace of the weak trace class operators contains the trace class operators. The
1730:{\displaystyle \left\{{\frac {a_{1}+a_{2}+\cdots +a_{n}}{n}}\right\}_{n=1}^{\infty }\in \ell _{1}}
1582:) and the commutator subspace of the finite rank operators are equal, ker Tr = Com(
1046:
2758:
2647:
2525:
2490:
66:
44:. This explicit spectral characterisation reduces problems and questions about commutators and
1446:{\displaystyle \left\{{\frac {a_{1}+a_{2}+\cdots +a_{n}}{n}}\right\}_{n=1}^{\infty }\in c_{00}}
625:{\displaystyle \prod _{k=0}^{n}\mu (k,A)\leq \prod _{k=0}^{n}\mu (k,B),\quad n=0,1,2,\ldots .}
165:
37:
2639:
2482:
1607:
1319:
40:
and it involves the invariance of the Calkin sequence space of an operator ideal to taking
234:. The following theorem is a slight extension to differences of normal operators (setting
8:
21:
2609:
K. Dykema; N. J. Kalton (1998). "Spectral characterization of sums of commutators, II".
2591:
2573:
2534:
2427:
2392:
2324:
1222:
1003:
858:, respectively, rearranged so that the absolute value of the eigenvalues is decreasing.
2661:
2504:
2735:
2705:
2685:
2595:
2463:
2446:
2431:
2396:
48:
on two-sided ideals to (more resolvable) problems and conditions on sequence spaces.
1932:
which has sum zero but does not have a summable sequence of Cesàro means. Hence Com(
1571:
2727:
2656:
2583:
2544:
2499:
2458:
2419:
2384:
2355:
2316:
169:
82:
57:
33:
25:
200:
180:
2704:, De Gruyter: Proceedings in Mathematics, Berlin: De Gruyter, pp. 311–332,
2721:
1617:
1022:
875:
is a sum of commutators if and only if the corresponding Calkin sequence space
238: = 0 in the following gives the statement of the previous sentence).
78:
45:
2549:
2520:
1279:
880:
223:
41:
2752:
2682:
Recent
Advances in Operator Theory, Operator Algebras, and their Applications
2360:
2206:{\displaystyle \left\{a_{1}+a_{2}+\cdots +a_{n}\right\}_{n=1}^{\infty }=O(1)}
29:
2564:
N. J. Kalton (1998). "Spectral characterization of sums of commutators, I".
74:
2731:
2587:
61:
2518:
2423:
2388:
2328:
455:
2637:
2578:
2480:
2320:
1964:
1329:) correspond to the space of sequences with finite non-zero terms,
2539:
2719:
2375:
G. Weiss (1980). "Commutators of
Hilbert–Schmidt Operators, II".
1192:
in an arbitrary orthonormal basis of the separable
Hilbert space
1127:
in an arbitrary orthonormal basis of the separable
Hilbert space
987:
in an arbitrary orthonormal basis of the separable Hilbert space
938:
in an arbitrary orthonormal basis of the separable Hilbert space
2410:
G. Weiss (1986). "Commutators of Hilbert–Schmidt Operators, I".
1835:{\displaystyle a_{n}={\frac {1}{n\log ^{2}(n)}},\quad n\geq 2.}
249:
are compact normal operators that belong to a two-sided ideal
2521:"Traces of compact operators and the noncommutative residue"
1271:) correspond to the space of converging to zero sequences,
2519:
N. J. Kalton; S. Lord; D. Potapov; F. Sukochev (2013).
865:
2608:
2117:
1990:
1862:
1771:
1636:
1473:
1352:
1049:
685:
506:
277:
2638:
K. Dykema; T. Figiel; G. Weiss; M. Wodzicki (2004).
2481:
K. Dykema; T. Figiel; G. Weiss; M. Wodzicki (2004).
850:) are the sequence of eigenvalues of the operators
2341:
1253:is a separable infinite dimensional Hilbert space.
899:, where C denotes the Cesàro operator on sequences.
2699:
2307:P. Halmos (1954). "Commutators of operators. II".
2205:
2089:
1916:{\displaystyle a_{1}=-\sum _{n=2}^{\infty }a_{n}.}
1915:
1834:
1729:
1524:
1445:
1057:
810:
624:
478:is a two-sided ideal such that a bounded operator
402:
2750:
2344:"On commutators in ideals of compact operators"
830:is the Calkin sequence space corresponding to
422:is the Calkin sequence space corresponding to
93:The commutator subspace of a two-sided ideal
2563:
2444:
159:
1525:{\displaystyle a_{1}+a_{2}+\cdots +a_{N}=0}
203:that belong to the commutator subspace Com(
2720:S. Lord, F. A. Sukochev. D. Zanin (2012).
2660:
2640:"Commutator structure of operator ideals"
2577:
2548:
2538:
2503:
2483:"Commutator structure of operator ideals"
2462:
2359:
2306:
1759:... = 0. An example is the sequence with
1051:
2723:Singular traces: theory and applications
2671:
2409:
2374:
1278:. For a converging to zero sequence the
1016:
2447:"Trace-class operators and commutators"
669:belongs to the commutator subspace Com(
261:belongs to the commutator subspace Com(
2751:
2476:
2474:
2412:Integral Equations and Operator Theory
2377:Integral Equations and Operator Theory
1077:has a non-zero trace if and only if C(
486:whenever there is a bounded operator
191:belongs to the Calkin sequence space
1745:is stronger than the condition that
866:Consequences of the characterisation
497:
462:belong to the Calkin sequence space
2621:
2512:
2471:
1207:In any two-sided ideal satisfying (
1146:In any two-sided ideal satisfying (
945:In any two-sided ideal satisfying (
109:is the linear span of operators in
13:
2438:
2403:
2368:
2300:
2183:
2082:
2063:
1895:
1709:
1425:
1135:are in direct correspondence with
797:
389:
32:is the linear subspace spanned by
14:
2775:
2602:
2557:
2335:
1939:) ⊊ ker Tr ⊊
1069:). The consequences above imply
1006:in a two-sided ideal satisfying (
97:of the bounded linear operators
2309:American Journal of Mathematics
1822:
1282:converge to zero. Therefore, C(
1107:)) for every positive operator
1027:A trace φ on a two-sided ideal
591:
105:) on a separable Hilbert space
2451:Journal of Functional Analysis
2342:C. Pearcy; D. Topping (1971).
2200:
2194:
1813:
1807:
918:) for every positive operator
772:
760:
751:
739:
585:
573:
543:
531:
364:
352:
343:
331:
1:
2662:10.1016/s0001-8708(03)00141-5
2631:
2505:10.1016/s0001-8708(03)00141-5
2348:Michigan Mathematical Journal
1188:)) is the diagonalisation of
1123:)) is the diagonalisation of
983:)) is the diagonalisation of
934:)) is the diagonalisation of
442:) are the singular values of
207:) can characterised as those
175:belongs to a two-sided ideal
88:
2464:10.1016/0022-1236(89)90064-5
1204:) is an eigenvalue sequence.
1058:{\displaystyle \mathbb {C} }
999:) is an eigenvalue sequence.
7:
1244:
1209:
1148:
1008:
947:
638:
144:The commutator subspace of
10:
2780:
1955:weak trace class operators
1951:Weak trace class operators
1221:) = 0 for every
1020:
1012:) is a sum of commutators.
879:is invariant under taking
51:
2550:10.1016/j.aim.2012.11.007
2222:it is immediate that Com(
1039:is a linear functional φ:
160:Spectral characterisation
113:of the form =
71:Hilbert–Schmidt operators
2293:
1262:compact linear operators
1223:quasi-nilpotent operator
1004:quasi-nilpotent operator
955: − diag(
906: − diag(
649:If the bounded operator
148:is a linear subspace of
73:. British mathematician
26:bounded linear operators
2648:Advances in Mathematics
2526:Advances in Mathematics
2491:Advances in Mathematics
2726:. Berlin: De Gruyter.
2361:10.1307/mmj/1029000686
2207:
2091:
1917:
1899:
1836:
1731:
1604:Trace class operators.
1526:
1461:occurs if and only if
1447:
1316:Finite rank operators.
1172:)) for every operator
1059:
812:
730:
626:
569:
527:
404:
322:
2732:10.1515/9783110262551
2588:10.1515/crll.1998.102
2445:N. J. Kalton (1989).
2208:
2092:
1918:
1879:
1837:
1732:
1608:trace class operators
1527:
1448:
1320:finite rank operators
1137:symmetric functionals
1131:. That is, traces on
1065:that vanishes on Com(
1060:
1017:Application to traces
967:) for every operator
813:
710:
627:
549:
507:
405:
302:
166:Calkin correspondence
38:Calkin correspondence
2764:Von Neumann algebras
2611:J. Reine Angew. Math
2566:J. Reine Angew. Math
2115:
1988:
1974:. From the condition
1860:
1769:
1634:
1570:. The kernel of the
1471:
1350:
1073:The two-sided ideal
1047:
683:
504:
458:of all operators in
456:eigenvalue sequences
275:
16:In mathematics, the
2187:
2067:
1713:
1429:
891:is equivalent to C(
801:
665: −
393:
257: −
179:if and only if the
117: −
18:commutator subspace
2672:G. Weiss (2005), "
2424:10.1007/bf01202521
2389:10.1007/BF01702316
2203:
2118:
2087:
1991:
1963:correspond to the
1913:
1832:
1727:
1637:
1618:summable sequences
1616:correspond to the
1540:for the sequence (
1522:
1443:
1353:
1258:Compact operators.
1164: ∘ diag(
1099: ∘ diag(
1055:
963:)) belongs to Com(
914:)) belongs to Com(
883:. In symbols, Com(
871:Every operator in
808:
686:
622:
454:Provided that the
400:
278:
226:of the sequence μ(
121:for all operators
2741:978-3-11-026255-1
2711:978-3-11-019805-8
2691:978-3-7643-7127-2
2249:harmonic sequence
2045:
1817:
1691:
1563:, 0, 0 , ...) in
1407:
708:
673:) if and only if
646:
645:
300:
265:) if and only if
81:, Gary Weiss and
2771:
2745:
2714:
2694:
2666:
2664:
2644:
2625:
2619:
2618:
2606:
2600:
2599:
2581:
2572:(504): 115–125.
2561:
2555:
2554:
2552:
2542:
2516:
2510:
2509:
2507:
2487:
2478:
2469:
2468:
2466:
2442:
2436:
2435:
2407:
2401:
2400:
2372:
2366:
2365:
2363:
2339:
2333:
2332:
2304:
2255:,... belongs to
2251:1,1/2,1/3,...,1/
2212:
2210:
2209:
2204:
2186:
2181:
2170:
2166:
2165:
2164:
2146:
2145:
2133:
2132:
2105:or equivalently
2096:
2094:
2093:
2088:
2086:
2085:
2066:
2061:
2050:
2046:
2041:
2040:
2039:
2021:
2020:
2008:
2007:
1997:
1922:
1920:
1919:
1914:
1909:
1908:
1898:
1893:
1872:
1871:
1841:
1839:
1838:
1833:
1818:
1816:
1803:
1802:
1786:
1781:
1780:
1736:
1734:
1733:
1728:
1726:
1725:
1712:
1707:
1696:
1692:
1687:
1686:
1685:
1667:
1666:
1654:
1653:
1643:
1531:
1529:
1528:
1523:
1515:
1514:
1496:
1495:
1483:
1482:
1452:
1450:
1449:
1444:
1442:
1441:
1428:
1423:
1412:
1408:
1403:
1402:
1401:
1383:
1382:
1370:
1369:
1359:
1232:and every trace
1064:
1062:
1061:
1056:
1054:
817:
815:
814:
809:
800:
795:
784:
780:
779:
775:
729:
724:
709:
707:
693:
640:
631:
629:
628:
623:
568:
563:
526:
521:
498:
409:
407:
406:
401:
392:
387:
376:
372:
371:
367:
321:
316:
301:
299:
285:
201:Normal operators
170:compact operator
83:Mariusz Wodzicki
58:matrix mechanics
2779:
2778:
2774:
2773:
2772:
2770:
2769:
2768:
2749:
2748:
2742:
2712:
2692:
2642:
2634:
2629:
2628:
2622:
2607:
2603:
2562:
2558:
2517:
2513:
2485:
2479:
2472:
2443:
2439:
2408:
2404:
2373:
2369:
2340:
2336:
2321:10.2307/2372409
2305:
2301:
2296:
2289:
2282:
2275:
2268:
2261:
2246:
2242:
2235:
2231:
2182:
2171:
2160:
2156:
2141:
2137:
2128:
2124:
2123:
2119:
2116:
2113:
2112:
2075:
2071:
2062:
2051:
2035:
2031:
2016:
2012:
2003:
1999:
1998:
1996:
1992:
1989:
1986:
1985:
1971:
1962:
1945:
1938:
1904:
1900:
1894:
1883:
1867:
1863:
1861:
1858:
1857:
1798:
1794:
1790:
1785:
1776:
1772:
1770:
1767:
1766:
1758:
1751:
1721:
1717:
1708:
1697:
1681:
1677:
1662:
1658:
1649:
1645:
1644:
1642:
1638:
1635:
1632:
1631:
1620:. The condition
1615:
1590:)) ⊊
1569:
1562:
1553:
1546:
1510:
1506:
1491:
1487:
1478:
1474:
1472:
1469:
1468:
1437:
1433:
1424:
1413:
1397:
1393:
1378:
1374:
1365:
1361:
1360:
1358:
1354:
1351:
1348:
1347:
1336:. The condition
1335:
1304:)) =
1295:
1288:
1277:
1247:
1050:
1048:
1045:
1044:
1025:
1019:
868:
796:
785:
735:
731:
725:
714:
697:
692:
691:
687:
684:
681:
680:
564:
553:
522:
511:
505:
502:
501:
450:, respectively.
388:
377:
327:
323:
317:
306:
289:
284:
283:
279:
276:
273:
272:
181:singular values
162:
152:denoted by Com(
91:
67:Schatten ideals
54:
28:on a separable
20:of a two-sided
12:
11:
5:
2777:
2767:
2766:
2761:
2759:Hilbert spaces
2747:
2746:
2740:
2716:
2715:
2710:
2696:
2695:
2690:
2668:
2667:
2633:
2630:
2627:
2626:
2620:
2601:
2556:
2511:
2470:
2437:
2418:(6): 877–892.
2402:
2383:(4): 574–600.
2367:
2354:(3): 247–252.
2334:
2315:(1): 191–198.
2298:
2297:
2295:
2292:
2287:
2280:
2273:
2269:. In summary,
2266:
2259:
2244:
2240:
2236: = (
2233:
2226:
2220:
2219:
2218:
2217:
2216:
2215:
2214:
2213:
2202:
2199:
2196:
2193:
2190:
2185:
2180:
2177:
2174:
2169:
2163:
2159:
2155:
2152:
2149:
2144:
2140:
2136:
2131:
2127:
2122:
2103:
2102:
2101:
2100:
2099:
2098:
2097:
2084:
2081:
2078:
2074:
2070:
2065:
2060:
2057:
2054:
2049:
2044:
2038:
2034:
2030:
2027:
2024:
2019:
2015:
2011:
2006:
2002:
1995:
1976:
1975:
1972:sequence space
1969:
1960:
1943:
1936:
1930:
1929:
1928:
1927:
1926:
1925:
1924:
1923:
1912:
1907:
1903:
1897:
1892:
1889:
1886:
1882:
1878:
1875:
1870:
1866:
1848:
1847:
1846:
1845:
1844:
1843:
1842:
1831:
1828:
1825:
1821:
1815:
1812:
1809:
1806:
1801:
1797:
1793:
1789:
1784:
1779:
1775:
1756:
1749:
1743:
1742:
1741:
1740:
1739:
1738:
1737:
1724:
1720:
1716:
1711:
1706:
1703:
1700:
1695:
1690:
1684:
1680:
1676:
1673:
1670:
1665:
1661:
1657:
1652:
1648:
1641:
1622:
1621:
1613:
1600:
1599:
1572:operator trace
1567:
1558:
1551:
1544:
1538:
1537:
1536:
1535:
1534:
1533:
1532:
1521:
1518:
1513:
1509:
1505:
1502:
1499:
1494:
1490:
1486:
1481:
1477:
1459:
1458:
1457:
1456:
1455:
1454:
1453:
1440:
1436:
1432:
1427:
1422:
1419:
1416:
1411:
1406:
1400:
1396:
1392:
1389:
1386:
1381:
1377:
1373:
1368:
1364:
1357:
1338:
1337:
1333:
1313:
1293:
1286:
1275:
1246:
1243:
1242:
1241:
1205:
1160:) =
1144:
1086:
1081:) ≠
1053:
1023:Singular trace
1021:Main article:
1018:
1015:
1014:
1013:
1000:
943:
900:
895:) =
887:) =
867:
864:
860:
859:
824:
823:
822:
821:
820:
819:
818:
807:
804:
799:
794:
791:
788:
783:
778:
774:
771:
768:
765:
762:
759:
756:
753:
750:
747:
744:
741:
738:
734:
728:
723:
720:
717:
713:
706:
703:
700:
696:
690:
644:
643:
634:
632:
621:
618:
615:
612:
609:
606:
603:
600:
597:
594:
590:
587:
584:
581:
578:
575:
572:
567:
562:
559:
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195:associated to
168:states that a
161:
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90:
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79:Tadeusz Figiel
53:
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2:
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1993:
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1968:
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229:
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30:Hilbert space
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2701:
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2579:math/9709209
2569:
2565:
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2530:
2524:
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1330:
1326:
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1309:
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1301:
1297:
1290:
1283:
1280:Cesàro means
1272:
1268:
1264:
1257:
1250:
1248:
1237:
1233:
1229:
1225:
1218:
1214:
1208:
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1173:
1169:
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1116:
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1100:
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1092:
1088:
1082:
1078:
1074:
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1040:
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1032:
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1026:
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988:
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972:
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960:
956:
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935:
931:
927:
923:
919:
915:
911:
907:
903:
896:
892:
888:
884:
881:Cesàro means
876:
872:
861:
855:
851:
847:
843:
839:
835:
831:
827:
670:
666:
662:
658:
654:
650:
636:
491:
487:
483:
479:
475:
471:
463:
459:
453:
447:
443:
439:
435:
431:
427:
423:
419:
262:
258:
254:
250:
246:
242:
235:
231:
227:
219:
216:
212:
211:such that μ(
208:
204:
196:
192:
188:
184:
176:
172:
163:
153:
149:
145:
143:
138:
134:
130:
126:
122:
118:
114:
110:
106:
102:
98:
94:
92:
75:Nigel Kalton
55:
42:Cesàro means
17:
15:
2276:⊊ Com(
1180:where diag(
1115:where diag(
975:where diag(
926:where diag(
482:belongs to
224:Cesàro mean
62:Paul Halmos
34:commutators
2753:Categories
2632:References
2617:: 127–137.
657:belong to
89:Definition
2596:119124949
2540:1210.3423
2457:: 41–74.
2432:122936389
2397:189875793
2283:) ⊊
2184:∞
2151:⋯
2083:∞
2073:ℓ
2069:∈
2064:∞
2026:⋯
1896:∞
1881:∑
1877:−
1827:≥
1805:
1719:ℓ
1715:∈
1710:∞
1672:⋯
1501:⋯
1431:∈
1426:∞
1388:⋯
803:∈
798:∞
758:λ
755:−
737:λ
712:∑
617:…
571:μ
551:∏
547:≤
529:μ
509:∏
494:such that
395:∈
390:∞
350:μ
347:−
329:μ
304:∑
2655:: 1–79.
2533:: 1–55.
2498:: 1–79.
1554:, ... ,
1296:and Com(
1249:Suppose
1245:Examples
474:Suppose
472:Theorem.
253:. Then
245:Suppose
243:Theorem.
2329:2372409
156:) or .
52:History
2738:
2708:
2688:
2594:
2430:
2395:
2327:
1953:. The
1574:Tr on
1002:Every
826:where
418:where
46:traces
2643:(PDF)
2592:S2CID
2574:arXiv
2535:arXiv
2486:(PDF)
2428:S2CID
2393:S2CID
2325:JSTOR
2294:Notes
1965:weak-
1228:from
661:then
187:) of
133:from
125:from
22:ideal
2736:ISBN
2706:ISBN
2686:ISBN
2570:1998
1850:and
1606:The
1318:The
1289:) =
1260:The
1196:and
1095:) =
991:and
854:and
834:and
653:and
446:and
426:and
222:the
164:The
129:and
2728:doi
2657:doi
2653:185
2615:504
2584:doi
2545:doi
2531:235
2500:doi
2496:185
2459:doi
2420:doi
2385:doi
2356:doi
2317:doi
2288:1,∞
2281:1,∞
2267:1,∞
2260:1,∞
1961:1,∞
1796:log
1236:on
1213:),
1176:in
1152:),
1139:on
1111:in
1031:of
971:in
922:in
842:),
490:in
434:),
247:A,B
220:and
199:.
141:).
24:of
2755::
2734:.
2651:.
2645:.
2613:.
2590:.
2582:.
2568:.
2543:.
2529:.
2523:.
2494:.
2488:.
2473:^
2455:86
2453:.
2449:.
2426:.
2414:.
2391:.
2379:.
2352:18
2350:.
2346:.
2323:.
2313:76
2311:.
2290:.
2230:,∞
1946:.
1830:2.
1752:+
1598:).
1568:00
1547:,
1439:00
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1312:).
1043:→
1037:H)
183:μ(
119:BA
115:AB
2744:.
2730::
2678:H
2676:(
2674:B
2665:.
2659::
2598:.
2586::
2576::
2553:.
2547::
2537::
2508:.
2502::
2467:.
2461::
2434:.
2422::
2416:9
2399:.
2387::
2381:3
2364:.
2358::
2331:.
2319::
2285:L
2278:L
2274:1
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2264:l
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2234:+
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2201:)
2198:1
2195:(
2192:O
2189:=
2179:1
2176:=
2173:n
2168:}
2162:n
2158:a
2154:+
2148:+
2143:2
2139:a
2135:+
2130:1
2126:a
2121:{
2080:,
2077:1
2059:1
2056:=
2053:n
2048:}
2043:n
2037:n
2033:a
2029:+
2023:+
2018:2
2014:a
2010:+
2005:1
2001:a
1994:{
1970:1
1967:l
1958:L
1944:1
1941:L
1937:1
1934:L
1911:.
1906:n
1902:a
1891:2
1888:=
1885:n
1874:=
1869:1
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1824:n
1820:,
1814:)
1811:n
1808:(
1800:2
1792:n
1788:1
1783:=
1778:n
1774:a
1757:2
1754:a
1750:1
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1723:1
1705:1
1702:=
1699:n
1694:}
1689:n
1683:n
1679:a
1675:+
1669:+
1664:2
1660:a
1656:+
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1640:{
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1596:H
1594:(
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1586:(
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1578:(
1576:F
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1520:0
1517:=
1512:N
1508:a
1504:+
1498:+
1493:2
1489:a
1485:+
1480:1
1476:a
1435:c
1421:1
1418:=
1415:n
1410:}
1405:n
1399:n
1395:a
1391:+
1385:+
1380:2
1376:a
1372:+
1367:1
1363:a
1356:{
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1327:H
1325:(
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1308:(
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1300:(
1298:K
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1287:0
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1269:H
1267:(
1265:K
1251:H
1240:.
1238:J
1234:φ
1230:J
1226:Q
1219:Q
1217:(
1215:φ
1210:1
1202:A
1200:(
1198:λ
1194:H
1190:A
1186:A
1184:(
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1178:J
1174:A
1170:A
1168:(
1166:λ
1162:φ
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1156:(
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1143:.
1141:j
1133:J
1129:H
1125:A
1121:A
1119:(
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1109:A
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1103:(
1101:μ
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1091:(
1089:φ
1085:.
1083:j
1079:j
1075:J
1067:J
1052:C
1041:J
1035:(
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997:A
995:(
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989:H
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979:(
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973:J
969:A
965:J
961:A
959:(
957:λ
953:A
948:1
942:.
940:H
936:A
932:A
930:(
928:μ
924:J
920:A
916:J
912:A
910:(
908:μ
904:A
897:j
893:j
889:J
885:J
877:j
873:J
856:B
852:A
848:B
846:(
844:λ
840:A
838:(
836:λ
832:J
828:j
806:j
793:0
790:=
787:n
782:}
777:)
773:)
770:B
767:,
764:k
761:(
752:)
749:A
746:,
743:k
740:(
733:(
727:n
722:0
719:=
716:k
705:n
702:+
699:1
695:1
689:{
671:J
667:B
663:A
659:J
655:B
651:A
641:)
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637:(
620:.
614:,
611:2
608:,
605:1
602:,
599:0
596:=
593:n
589:,
586:)
583:B
580:,
577:k
574:(
566:n
561:0
558:=
555:k
544:)
541:A
538:,
535:k
532:(
524:n
519:0
516:=
513:k
492:J
488:B
484:J
480:A
476:J
464:j
460:J
448:B
444:A
440:B
438:(
436:μ
432:A
430:(
428:μ
424:J
420:j
398:j
385:0
382:=
379:n
374:}
369:)
365:)
362:B
359:,
356:k
353:(
344:)
341:A
338:,
335:k
332:(
325:(
319:n
314:0
311:=
308:k
297:n
294:+
291:1
287:1
281:{
263:J
259:B
255:A
251:J
236:B
232:j
228:A
217:j
213:A
209:A
205:J
197:J
193:j
189:A
185:A
177:J
173:A
154:J
150:J
146:J
139:H
137:(
135:B
131:B
127:J
123:A
111:J
107:H
103:H
101:(
99:B
95:J
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