1197:
320:
526:
1505:
933:
763:
1405:
1453:
570:
145:
944:
1538:
1262:
1360:
419:
214:
1334:
1297:
1558:
1593:
787:
is a generalization of the first resolvent identity, above, useful for comparing the resolvents of two distinct operators. Given operators
233:
438:
1733:
1462:
824:
654:
1777:
1755:
1676:
1373:
1410:
1563:
54:
24:
1800:
182:
42:
1805:
1598:
1192:{\displaystyle (A-zI)^{-1}-(B-zI)^{-1}=(A-zI)^{-1}((B-zI)-(A-zI))(B-zI)^{-1}=(A-zI)^{-1}(B-A)(B-zI)^{-1}\,.}
1588:
585:
534:
155:
78:
225:
151:
1510:
53:
and more general spaces. Formal justification for the manipulations can be found in the framework of
1795:
1232:
341:
65:
1343:
382:
192:
1310:
46:
361:
1267:
769:
8:
349:
326:
20:
1685:
1541:
1208:
1689:
1773:
1765:
1751:
1729:
1721:
1672:
1664:
1578:
430:
345:
1704:
1573:
1300:
166:
38:
1583:
1568:
597:
64:
captures the spectral properties of an operator in the analytic structure of the
1660:
1789:
1553:
1223:
637:
608:
589:
1367:
50:
1717:
30:
1743:
1709:
333:
315:{\displaystyle -{\frac {1}{2\pi i}}\oint _{C_{\lambda }}(A-zI)^{-1}~dz}
178:
150:
Among other uses, the resolvent may be used to solve the inhomogeneous
780:, instead, so that the formula above differs in sign from theirs.)
580:
The first major use of the resolvent operator as a series in
521:{\displaystyle R(z;A)=\int _{0}^{\infty }e^{-zt}U(t)~dt,}
379:
is a one-parameter group of unitary operators. Whenever
189:. That is, suppose there exists a simple closed curve
1513:
1465:
1413:
1376:
1346:
1313:
1270:
1235:
947:
827:
657:
537:
441:
385:
236:
195:
165:
can be used to directly obtain information about the
154:; a commonly used approach is a series solution, the
81:
1500:{\displaystyle \{\lambda _{i}\}_{i\in \mathbb {N} }}
1532:
1499:
1447:
1399:
1354:
1328:
1291:
1256:
1191:
928:{\displaystyle R(z;A)-R(z;B)=R(z;A)(B-A)R(z;B)\,.}
927:
758:{\displaystyle R(z;A)-R(w;A)=(z-w)R(z;A)R(w;A)\,.}
757:
564:
520:
413:
314:
208:
139:
1787:
1750:(2nd ed.), New York, NY: Springer-Verlag,
1659:
1594:Decomposition of spectrum (functional analysis)
1559:Stone's theorem on one-parameter unitary groups
1400:{\displaystyle \sigma (A)\subset \mathbb {R} }
795:, both defined on the same linear space, and
1728:, Providence: American Mathematical Society,
1716:
1634:Dunford and Schwartz, Vol I, Lemma 6, p. 568.
1448:{\displaystyle \{v_{i}\}_{i\in \mathbb {N} }}
1527:
1514:
1480:
1466:
1428:
1414:
408:
402:
1690:"Sur une classe d'equations fonctionnelles"
1625:Hille and Phillips, Theorem 11.4.1, p. 341
531:where the integral is taken along the ray
37:is a technique for applying concepts from
1708:
1643:Hille and Phillips, Theorem 4.8.2, p. 126
1491:
1439:
1393:
1348:
1185:
921:
751:
1748:Perturbation Theory for Linear Operators
1684:
648:(also called Hilbert's identity) holds:
1669:Linear Operators, Part I General Theory
1788:
1764:
1407:and there exists an orthonormal basis
348:to an integral over the one-parameter
216:in the complex plane that separates
614:
565:{\displaystyle \arg t=-\arg \lambda }
1742:
1307:has compact resolvent. The spectrum
1202:
140:{\displaystyle R(z;A)=(A-zI)^{-1}~.}
1726:Functional Analysis and Semi-groups
1671:, Hoboken, NJ: Wiley-Interscience,
13:
938:A one-line proof goes as follows:
473:
72:, the resolvent may be defined as
14:
1817:
1772:, New York, NY: Springer-Verlag,
772:, cited, define the resolvent as
220:from the rest of the spectrum of
1770:Partial Differential Equations I
1616:Taylor, section 9 of Appendix A.
1533:{\displaystyle \{\lambda _{i}\}}
352:of transformations generated by
344:relates the resolvent through a
1564:Holomorphic functional calculus
55:holomorphic functional calculus
25:Holomorphic functional calculus
1646:
1637:
1628:
1619:
1610:
1386:
1380:
1323:
1317:
1286:
1274:
1251:
1245:
1173:
1157:
1154:
1142:
1130:
1114:
1099:
1083:
1080:
1077:
1062:
1056:
1041:
1038:
1026:
1010:
995:
979:
964:
948:
918:
906:
900:
888:
885:
873:
864:
852:
843:
831:
818:the following identity holds,
748:
736:
730:
718:
712:
700:
694:
682:
673:
661:
592:, in a landmark 1903 paper in
503:
497:
457:
445:
395:
387:
291:
275:
119:
103:
97:
85:
1:
1604:
1599:Limiting absorption principle
1257:{\displaystyle z\in \rho (A)}
596:that helped establish modern
1355:{\displaystyle \mathbb {C} }
414:{\displaystyle |z|>\|A\|}
209:{\displaystyle C_{\lambda }}
7:
1547:
152:Fredholm integral equations
10:
1822:
1329:{\displaystyle \sigma (A)}
575:
18:
785:second resolvent identity
356:. Thus, for example, if
1589:LiouvilleāNeumann series
1340:is a discrete subset of
646:first resolvent identity
586:LiouvilleāNeumann series
429:can be expressed as the
156:LiouvilleāNeumann series
16:Technique in mathematics
1207:When studying a closed
173:. For example, suppose
1534:
1501:
1449:
1401:
1356:
1330:
1293:
1292:{\displaystyle R(z;A)}
1258:
1193:
929:
759:
566:
522:
415:
316:
210:
167:spectral decomposition
141:
1801:Formalism (deductive)
1535:
1502:
1450:
1402:
1357:
1331:
1294:
1259:
1194:
930:
760:
567:
523:
416:
362:skew-Hermitian matrix
317:
211:
142:
19:Further information:
1806:Mathematical physics
1511:
1507:respectively. Also,
1463:
1455:of eigenvectors of
1411:
1374:
1344:
1311:
1268:
1233:
945:
825:
770:Dunford and Schwartz
655:
535:
439:
383:
342:HilleāYosida theorem
234:
193:
79:
68:. Given an operator
41:to the study of the
644:, we have that the
477:
421:, the resolvent of
327:projection operator
35:resolvent formalism
21:Frobenius covariant
1766:Taylor, Michael E.
1722:Phillips, Ralph S.
1710:10.1007/bf02421317
1665:Schwartz, Jacob T.
1542:accumulation point
1530:
1497:
1445:
1397:
1352:
1326:
1289:
1254:
1229:, if there exists
1209:unbounded operator
1189:
925:
755:
615:Resolvent identity
562:
518:
463:
411:
312:
206:
137:
1735:978-0-8218-1031-6
1686:Fredholm, Erik I.
1579:Laplace transform
1459:with eigenvalues
1362:. If furthermore
1203:Compact resolvent
508:
431:Laplace transform
346:Laplace transform
305:
256:
161:The resolvent of
133:
1813:
1782:
1760:
1738:
1713:
1712:
1697:Acta Mathematica
1694:
1681:
1653:
1650:
1644:
1641:
1635:
1632:
1626:
1623:
1617:
1614:
1574:Compact operator
1539:
1537:
1536:
1531:
1526:
1525:
1506:
1504:
1503:
1498:
1496:
1495:
1494:
1478:
1477:
1458:
1454:
1452:
1451:
1446:
1444:
1443:
1442:
1426:
1425:
1406:
1404:
1403:
1398:
1396:
1365:
1361:
1359:
1358:
1353:
1351:
1339:
1335:
1333:
1332:
1327:
1306:
1301:compact operator
1298:
1296:
1295:
1290:
1263:
1261:
1260:
1255:
1228:
1221:
1217:
1213:
1198:
1196:
1195:
1190:
1184:
1183:
1141:
1140:
1110:
1109:
1037:
1036:
1006:
1005:
975:
974:
934:
932:
931:
926:
817:
798:
794:
790:
779:
764:
762:
761:
756:
643:
640:of an operator
635:
624:
594:Acta Mathematica
583:
571:
569:
568:
563:
527:
525:
524:
519:
506:
493:
492:
476:
471:
420:
418:
417:
412:
398:
390:
378:
359:
355:
339:
332:
321:
319:
318:
313:
303:
302:
301:
274:
273:
272:
271:
257:
255:
241:
223:
219:
215:
213:
212:
207:
205:
204:
188:
176:
172:
164:
146:
144:
143:
138:
131:
130:
129:
71:
39:complex analysis
1821:
1820:
1816:
1815:
1814:
1812:
1811:
1810:
1796:Fredholm theory
1786:
1785:
1780:
1758:
1736:
1692:
1679:
1661:Dunford, Nelson
1656:
1652:Taylor, p. 515.
1651:
1647:
1642:
1638:
1633:
1629:
1624:
1620:
1615:
1611:
1607:
1584:Fredholm theory
1569:Spectral theory
1550:
1521:
1517:
1512:
1509:
1508:
1490:
1483:
1479:
1473:
1469:
1464:
1461:
1460:
1456:
1438:
1431:
1427:
1421:
1417:
1412:
1409:
1408:
1392:
1375:
1372:
1371:
1363:
1347:
1345:
1342:
1341:
1337:
1312:
1309:
1308:
1304:
1303:, we say that
1269:
1266:
1265:
1234:
1231:
1230:
1226:
1219:
1215:
1211:
1205:
1176:
1172:
1133:
1129:
1102:
1098:
1029:
1025:
998:
994:
967:
963:
946:
943:
942:
826:
823:
822:
800:
796:
792:
788:
773:
656:
653:
652:
641:
626:
620:
617:
598:operator theory
581:
578:
536:
533:
532:
482:
478:
472:
467:
440:
437:
436:
394:
386:
384:
381:
380:
365:
357:
353:
337:
330:
294:
290:
267:
263:
262:
258:
245:
240:
235:
232:
231:
221:
217:
200:
196:
194:
191:
190:
186:
177:is an isolated
174:
170:
162:
122:
118:
80:
77:
76:
69:
27:
17:
12:
11:
5:
1819:
1809:
1808:
1803:
1798:
1784:
1783:
1778:
1762:
1756:
1740:
1734:
1714:
1682:
1677:
1655:
1654:
1645:
1636:
1627:
1618:
1608:
1606:
1603:
1602:
1601:
1596:
1591:
1586:
1581:
1576:
1571:
1566:
1561:
1556:
1549:
1546:
1540:has no finite
1529:
1524:
1520:
1516:
1493:
1489:
1486:
1482:
1476:
1472:
1468:
1441:
1437:
1434:
1430:
1424:
1420:
1416:
1395:
1391:
1388:
1385:
1382:
1379:
1350:
1325:
1322:
1319:
1316:
1288:
1285:
1282:
1279:
1276:
1273:
1253:
1250:
1247:
1244:
1241:
1238:
1204:
1201:
1200:
1199:
1188:
1182:
1179:
1175:
1171:
1168:
1165:
1162:
1159:
1156:
1153:
1150:
1147:
1144:
1139:
1136:
1132:
1128:
1125:
1122:
1119:
1116:
1113:
1108:
1105:
1101:
1097:
1094:
1091:
1088:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1052:
1049:
1046:
1043:
1040:
1035:
1032:
1028:
1024:
1021:
1018:
1015:
1012:
1009:
1004:
1001:
997:
993:
990:
987:
984:
981:
978:
973:
970:
966:
962:
959:
956:
953:
950:
936:
935:
924:
920:
917:
914:
911:
908:
905:
902:
899:
896:
893:
890:
887:
884:
881:
878:
875:
872:
869:
866:
863:
860:
857:
854:
851:
848:
845:
842:
839:
836:
833:
830:
808:) ā©
766:
765:
754:
750:
747:
744:
741:
738:
735:
732:
729:
726:
723:
720:
717:
714:
711:
708:
705:
702:
699:
696:
693:
690:
687:
684:
681:
678:
675:
672:
669:
666:
663:
660:
616:
613:
577:
574:
561:
558:
555:
552:
549:
546:
543:
540:
529:
528:
517:
514:
511:
505:
502:
499:
496:
491:
488:
485:
481:
475:
470:
466:
462:
459:
456:
453:
450:
447:
444:
410:
407:
404:
401:
397:
393:
389:
323:
322:
311:
308:
300:
297:
293:
289:
286:
283:
280:
277:
270:
266:
261:
254:
251:
248:
244:
239:
203:
199:
148:
147:
136:
128:
125:
121:
117:
114:
111:
108:
105:
102:
99:
96:
93:
90:
87:
84:
15:
9:
6:
4:
3:
2:
1818:
1807:
1804:
1802:
1799:
1797:
1794:
1793:
1791:
1781:
1779:7-5062-4252-4
1775:
1771:
1767:
1763:
1759:
1757:0-387-07558-5
1753:
1749:
1745:
1741:
1737:
1731:
1727:
1723:
1719:
1715:
1711:
1706:
1702:
1698:
1691:
1687:
1683:
1680:
1678:0-471-60848-3
1674:
1670:
1666:
1662:
1658:
1657:
1649:
1640:
1631:
1622:
1613:
1609:
1600:
1597:
1595:
1592:
1590:
1587:
1585:
1582:
1580:
1577:
1575:
1572:
1570:
1567:
1565:
1562:
1560:
1557:
1555:
1554:Resolvent set
1552:
1551:
1545:
1543:
1522:
1518:
1487:
1484:
1474:
1470:
1435:
1432:
1422:
1418:
1389:
1383:
1377:
1369:
1320:
1314:
1302:
1283:
1280:
1277:
1271:
1248:
1242:
1239:
1236:
1225:
1224:Hilbert space
1210:
1186:
1180:
1177:
1169:
1166:
1163:
1160:
1151:
1148:
1145:
1137:
1134:
1126:
1123:
1120:
1117:
1111:
1106:
1103:
1095:
1092:
1089:
1086:
1074:
1071:
1068:
1065:
1059:
1053:
1050:
1047:
1044:
1033:
1030:
1022:
1019:
1016:
1013:
1007:
1002:
999:
991:
988:
985:
982:
976:
971:
968:
960:
957:
954:
951:
941:
940:
939:
922:
915:
912:
909:
903:
897:
894:
891:
882:
879:
876:
870:
867:
861:
858:
855:
849:
846:
840:
837:
834:
828:
821:
820:
819:
815:
811:
807:
803:
786:
781:
777:
771:
752:
745:
742:
739:
733:
727:
724:
721:
715:
709:
706:
703:
697:
691:
688:
685:
679:
676:
670:
667:
664:
658:
651:
650:
649:
647:
639:
638:resolvent set
633:
629:
623:
612:
610:
609:David Hilbert
607:was given by
606:
601:
599:
595:
591:
590:Ivar Fredholm
587:
573:
559:
556:
553:
550:
547:
544:
541:
538:
515:
512:
509:
500:
494:
489:
486:
483:
479:
468:
464:
460:
454:
451:
448:
442:
435:
434:
433:
432:
428:
424:
405:
399:
391:
376:
372:
368:
363:
351:
347:
343:
335:
328:
309:
306:
298:
295:
287:
284:
281:
278:
268:
264:
259:
252:
249:
246:
242:
237:
230:
229:
228:
227:
201:
197:
184:
180:
168:
159:
157:
153:
134:
126:
123:
115:
112:
109:
106:
100:
94:
91:
88:
82:
75:
74:
73:
67:
63:
58:
56:
52:
51:Banach spaces
48:
44:
40:
36:
32:
26:
22:
1769:
1747:
1725:
1718:Hille, Einar
1700:
1696:
1668:
1648:
1639:
1630:
1621:
1612:
1368:self-adjoint
1206:
937:
813:
809:
805:
801:
784:
782:
775:
767:
645:
631:
627:
621:
618:
604:
602:
593:
579:
530:
426:
422:
374:
370:
366:
324:
160:
149:
61:
59:
34:
28:
1744:Kato, Tosio
1703:: 365ā390,
768:(Note that
224:. Then the
31:mathematics
1790:Categories
1605:References
1264:such that
334:eigenspace
329:onto the
325:defines a
179:eigenvalue
66:functional
1519:λ
1488:∈
1471:λ
1436:∈
1390:⊂
1378:σ
1336:of such
1315:σ
1243:ρ
1240:∈
1178:−
1164:−
1149:−
1135:−
1121:−
1104:−
1090:−
1069:−
1060:−
1048:−
1031:−
1017:−
1000:−
986:−
977:−
969:−
955:−
895:−
847:−
707:−
677:−
605:resolvent
603:The name
588:) was by
560:λ
557:
551:−
542:
484:−
474:∞
465:∫
409:‖
403:‖
296:−
282:−
269:λ
260:∮
250:π
238:−
202:λ
124:−
110:−
62:resolvent
47:operators
1768:(1996),
1746:(1980),
1724:(1957),
1688:(1903),
1667:(1988),
1548:See also
619:For all
373:) = exp(
183:spectrum
181:in the
43:spectrum
1370:, then
576:History
364:, then
340:. The
226:residue
1776:
1754:
1732:
1675:
636:, the
507:
304:
132:
33:, the
1693:(PDF)
1299:is a
1222:on a
791:and
776:zI āA
625:in
584:(cf.
360:is a
350:group
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