1205:
2530:
1906:
654:
370:
969:
521:
1730:
204:
1756:
1135:
1960:). Property 2 and the continuous functional calculus ensure that Ί preserves the *-operation. Finally, the semicommutator property shows that Ί is multiplicative. Therefore the theorem holds.
532:
1057:
830:
1582:
1526:
258:
1638:
867:
2419:
410:
2082:
1976:
2245:
2805:
2372:
2227:
2566:
2203:
1667:
151:
2095:
1901:{\displaystyle \Phi :C^{*}(S)\rightarrow C^{*}(V)\quad {\text{by}}\quad \Phi (T_{f}+K)=\bigoplus _{\alpha \in A}(T_{f}+K)\oplus f(U).}
2690:
2184:
2075:
2051:
2030:
2009:
1376:
2454:
2099:
1736:
1140:
In this terminology, the Wold decomposition expresses an isometry as a direct sum of a pure isometry and a unitary operator.
987:
2638:
2250:
1089:
2705:
2306:
2795:
2643:
2533:
2255:
2240:
2068:
2270:
2559:
649:{\displaystyle H=\left(\bigoplus _{i\geq 0}M_{i}\right)\oplus \left(\bigcap _{i\geq 0}H_{i}\right)=K_{1}\oplus K_{2}.}
2515:
2275:
1455:), in the description of the Toeplitz algebra and that the spectrum of a unitary operator is contained in the circle
2746:
2659:
2469:
2393:
2510:
2326:
2766:
2260:
1004:
2790:
2695:
2362:
2163:
2235:
792:
365:{\displaystyle H=H\supset V(H)\supset V^{2}(H)\supset \cdots =H_{0}\supset H_{1}\supset H_{2}\supset \cdots ,}
1532:
2459:
1468:
2664:
2633:
2552:
2490:
2434:
2398:
1235:
The decomposition above can be generalized slightly to a sequence of isometries, indexed by the integers.
94:
59:
1589:
2715:
2669:
2612:
964:{\displaystyle V=V\vert _{K_{1}}\oplus V\vert _{K_{2}}=\left(\bigoplus _{\alpha \in A}S\right)\oplus U,}
27:
This article is about the general mathematical result. For the application to time series analysis, see
2800:
2598:
2473:
82:
2594:
2439:
2377:
2091:
93:
can be decomposed into a pair of uncorrelated processes, one deterministic, and the other being a
2464:
2331:
516:{\displaystyle M_{i}=H_{i}\ominus H_{i+1}=V^{i}(H\ominus V(H))\quad {\text{for}}\quad i\geq 0\;,}
90:
86:
2720:
2700:
2673:
2617:
2589:
2444:
2047:
2026:
2005:
1356:
28:
2730:
2607:
2449:
2367:
2336:
2316:
2301:
2296:
2291:
2128:
1984:
1417:
218:
75:
71:
55:
2710:
2311:
2265:
2213:
2208:
2179:
2060:
773:
39:
2138:
2500:
2352:
2153:
1204:
2784:
2603:
2575:
2505:
2429:
2158:
2143:
2133:
1148:
110:
67:
1989:
1971:
2495:
2148:
2118:
17:
2041:
2020:
1999:
990:
of any proper, i.e. non-unitary, isometry is the unit disk in the complex plane.
2725:
2424:
2414:
2321:
2123:
1952:= 0. Since the range of Ί is a C*-algebra, Ί is surjective by the minimality of
51:
35:
2357:
2197:
2193:
2189:
1264:
1911:
One can now verify Ί is an isomorphism that maps the unilateral shift to
63:
2004:. Operator Theory, Advances and Applications. Vol. 82. BirkhÀuser.
1725:{\displaystyle V=\left(\bigoplus _{\alpha \in A}T_{z}\right)\oplus U.}
2544:
1918:
By property 1 above, Ί is linear. The map Ί is injective because
1462:
The following properties of the
Toeplitz algebra will be needed:
199:{\displaystyle V=\left(\bigoplus _{\alpha \in A}S\right)\oplus U}
70:. It states that every isometry is a direct sum of copies of the
2548:
2064:
244:
A proof can be sketched as follows. Successive applications of
1199:
1287:. The Wold decomposition can be applied to characterize
1216:
1007:
232:
is a unitary operator (possible vacuous). The family {
2001:
Schur
Parameters, Factorization and Dilation Problems
1759:
1670:
1592:
1535:
1471:
1092:
986:
It is immediate from the Wold decomposition that the
870:
795:
535:
413:
261:
154:
1130:{\displaystyle V=\bigoplus _{1\leq \alpha \leq N}S.}
2739:
2683:
2652:
2626:
2582:
2483:
2407:
2386:
2345:
2284:
2226:
2172:
2107:
709:is a surjective isometry, i.e., a unitary operator
2420:Spectral theory of ordinary differential equations
1900:
1724:
1632:
1576:
1520:
1129:
1079:. In other words, a pure isometry of multiplicity
1051:
963:
824:
648:
515:
364:
198:
1306:) be the continuous functions on the unit circle
1188:defined above is a wandering subspace of
786:can be written as a direct sum Hilbert spaces
2560:
2076:
8:
2022:Banach Algebra Techniques in Operator Theory
1433:) is isomorphic to the Toeplitz algebra and
1043:
1037:
904:
881:
1052:{\textstyle \bigcap _{i\geq 0}H_{i}=\{0\}.}
2567:
2553:
2545:
2111:
2083:
2069:
2061:
2040:Rosenblum, Marvin; Rovnyak, James (1985).
509:
1988:
1865:
1846:
1824:
1808:
1792:
1770:
1758:
1702:
1686:
1669:
1629:
1620:
1607:
1597:
1591:
1559:
1558:
1545:
1540:
1534:
1517:
1502:
1489:
1476:
1470:
1447:The proof hinges on the connections with
1103:
1091:
1028:
1012:
1006:
932:
912:
907:
889:
884:
869:
816:
800:
794:
637:
624:
606:
590:
567:
551:
534:
494:
463:
444:
431:
418:
412:
347:
334:
321:
293:
260:
248:give a descending sequences of copies of
241:} consists of isomorphic Hilbert spaces.
170:
153:
2373:Group algebra of a locally compact group
1279:) is the norm closure of polynomials in
1071:, i.e. the cardinality of the index set
1001:if, in the notation of the above proof,
825:{\displaystyle K_{1}=\oplus H_{\alpha }}
1577:{\displaystyle T_{f}^{*}=T_{\bar {f}}.}
1239:The C*-algebra generated by an isometry
1521:{\displaystyle T_{f}+T_{g}=T_{f+g}.\,}
7:
1318:) generated by the unilateral shift
1633:{\displaystyle T_{f}T_{g}-T_{fg}\,}
252:isomorphically embedded in itself:
1814:
1760:
1067:is the dimension of the kernel of
25:
2691:Compact operator on Hilbert space
2043:Hardy Classes and Operator Theory
1644:The Wold decomposition says that
974:which is a Wold decomposition of
2529:
2528:
2455:Topological quantum field theory
1925:is not compact for any non-zero
1310:. We recall that the C*-algebra
1203:
765:. Suppose the dimension of each
2806:Theorems in functional analysis
1990:10.1090/S0002-9904-1967-11845-7
1972:"The C*-algebra of an isometry"
1813:
1807:
1648:is the direct sum of copies of
723:is isomorphic to another, with
499:
493:
85:, the theorem implies that any
1892:
1886:
1877:
1858:
1836:
1817:
1804:
1798:
1785:
1782:
1776:
1737:continuous functional calculus
1564:
490:
487:
481:
469:
305:
299:
283:
277:
121:) be the bounded operators on
48:Woldâvon Neumann decomposition
1:
2251:Uniform boundedness principle
1075:in the Wold decomposition of
842:is an invariant subspaces of
727:being an isomorphism between
1400:is the identity function in
2046:. Oxford University Press.
1998:Constantinescu, T. (1996).
1437:is the isomorphic image of
673:are invariant subspaces of
141:states that every isometry
2822:
2660:Hilbert projection theorem
2394:Invariant subspace problem
64:isometric linear operators
26:
2639:CauchyâSchwarz inequality
2524:
2114:
1322:takes the following form
2363:Spectrum of a C*-algebra
1383:In this identification,
1196:A sequence of isometries
857:is the unilateral shift
2460:Noncommutative geometry
2019:Douglas, R. G. (1972).
1359:with continuous symbol
383:) denotes the range of
2516:TomitaâTakesaki theory
2491:Approximation property
2435:Calculus of variations
1977:Bull. Amer. Math. Soc.
1902:
1726:
1657:and then some unitary
1634:
1578:
1522:
1179:. In particular, each
1131:
1053:
965:
826:
650:
517:
366:
200:
137:) be an isometry. The
95:moving average process
60:classification theorem
2670:Polarization identity
2613:Orthogonal complement
2511:BanachâMazur distance
2474:Generalized functions
1903:
1727:
1635:
1579:
1523:
1243:Consider an isometry
1132:
1054:
966:
827:
651:
518:
367:
201:
2644:Riesz representation
2599:L-semi-inner product
2256:Kakutani fixed-point
2241:Riesz representation
1757:
1668:
1590:
1533:
1469:
1090:
1005:
868:
793:
533:
411:
387:. The above defined
259:
152:
83:time series analysis
2796:Invariant subspaces
2665:Parseval's identity
2634:Bessel's inequality
2440:Functional calculus
2399:Mahler's conjecture
2378:Von Neumann algebra
2092:Functional analysis
1970:Coburn, L. (1967).
1586:The semicommutator
1550:
1063:of a pure isometry
850:restricted to each
221:on a Hilbert space
209:for some index set
2465:Riemann hypothesis
2164:Topological vector
2025:. Academic Press.
1898:
1857:
1722:
1697:
1630:
1574:
1536:
1518:
1215:. You can help by
1149:wandering subspace
1127:
1120:
1049:
1023:
961:
943:
822:
716:Furthermore, each
698:. In other words,
646:
601:
562:
513:
404:). If one defines
362:
196:
181:
139:Wold decomposition
91:stochastic process
44:Wold decomposition
38:, particularly in
18:Wold decomposition
2778:
2777:
2721:Sesquilinear form
2674:Parallelogram law
2618:Orthonormal basis
2542:
2541:
2445:Integral operator
2222:
2221:
1842:
1811:
1735:So we invoke the
1682:
1567:
1357:Toeplitz operator
1233:
1232:
1099:
1008:
928:
659:It is clear that
586:
547:
497:
166:
16:(Redirected from
2813:
2608:Prehilbert space
2569:
2562:
2555:
2546:
2532:
2531:
2450:Jones polynomial
2368:Operator algebra
2112:
2085:
2078:
2071:
2062:
2057:
2036:
2015:
1994:
1992:
1907:
1905:
1904:
1899:
1870:
1869:
1856:
1829:
1828:
1812:
1809:
1797:
1796:
1775:
1774:
1731:
1729:
1728:
1723:
1712:
1708:
1707:
1706:
1696:
1639:
1637:
1636:
1631:
1628:
1627:
1612:
1611:
1602:
1601:
1583:
1581:
1580:
1575:
1570:
1569:
1568:
1560:
1549:
1544:
1527:
1525:
1524:
1519:
1513:
1512:
1494:
1493:
1481:
1480:
1424:Theorem (Coburn)
1418:Toeplitz algebra
1416:) is called the
1377:compact operator
1228:
1225:
1207:
1200:
1136:
1134:
1133:
1128:
1119:
1058:
1056:
1055:
1050:
1033:
1032:
1022:
970:
968:
967:
962:
951:
947:
942:
919:
918:
917:
916:
896:
895:
894:
893:
831:
829:
828:
823:
821:
820:
805:
804:
655:
653:
652:
647:
642:
641:
629:
628:
616:
612:
611:
610:
600:
577:
573:
572:
571:
561:
522:
520:
519:
514:
498:
495:
468:
467:
455:
454:
436:
435:
423:
422:
371:
369:
368:
363:
352:
351:
339:
338:
326:
325:
298:
297:
219:unilateral shift
205:
203:
202:
197:
189:
185:
180:
76:unitary operator
72:unilateral shift
56:John von Neumann
21:
2821:
2820:
2816:
2815:
2814:
2812:
2811:
2810:
2791:Operator theory
2781:
2780:
2779:
2774:
2767:SegalâBargmann
2735:
2706:HilbertâSchmidt
2696:Densely defined
2679:
2648:
2622:
2578:
2573:
2543:
2538:
2520:
2484:Advanced topics
2479:
2403:
2382:
2341:
2307:HilbertâSchmidt
2280:
2271:GelfandâNaimark
2218:
2168:
2103:
2089:
2054:
2039:
2033:
2018:
2012:
1997:
1969:
1966:
1942:
1923:
1861:
1820:
1788:
1766:
1755:
1754:
1698:
1681:
1677:
1666:
1665:
1656:
1616:
1603:
1593:
1588:
1587:
1554:
1531:
1530:
1498:
1485:
1472:
1467:
1466:
1442:
1408:). The algebra
1395:
1354:
1341:
1241:
1229:
1223:
1220:
1213:needs expansion
1198:
1187:
1088:
1087:
1083:takes the form
1024:
1003:
1002:
984:
927:
923:
908:
903:
885:
880:
866:
865:
855:
840:
812:
796:
791:
790:
785:
779:. We see that
774:cardinal number
770:
764:
753:
743:
732:
721:
708:
697:
690:
672:
665:
633:
620:
602:
585:
581:
563:
546:
542:
531:
530:
459:
440:
427:
414:
409:
408:
395:
343:
330:
317:
289:
257:
256:
240:
226:
165:
161:
150:
149:
145:takes the form
103:
40:operator theory
32:
23:
22:
15:
12:
11:
5:
2819:
2817:
2809:
2808:
2803:
2798:
2793:
2783:
2782:
2776:
2775:
2773:
2772:
2764:
2758:compact &
2743:
2741:
2737:
2736:
2734:
2733:
2728:
2723:
2718:
2713:
2708:
2703:
2701:Hermitian form
2698:
2693:
2687:
2685:
2681:
2680:
2678:
2677:
2667:
2662:
2656:
2654:
2650:
2649:
2647:
2646:
2641:
2636:
2630:
2628:
2624:
2623:
2621:
2620:
2615:
2610:
2601:
2592:
2586:
2584:
2583:Basic concepts
2580:
2579:
2576:Hilbert spaces
2574:
2572:
2571:
2564:
2557:
2549:
2540:
2539:
2537:
2536:
2525:
2522:
2521:
2519:
2518:
2513:
2508:
2503:
2501:Choquet theory
2498:
2493:
2487:
2485:
2481:
2480:
2478:
2477:
2467:
2462:
2457:
2452:
2447:
2442:
2437:
2432:
2427:
2422:
2417:
2411:
2409:
2405:
2404:
2402:
2401:
2396:
2390:
2388:
2384:
2383:
2381:
2380:
2375:
2370:
2365:
2360:
2355:
2353:Banach algebra
2349:
2347:
2343:
2342:
2340:
2339:
2334:
2329:
2324:
2319:
2314:
2309:
2304:
2299:
2294:
2288:
2286:
2282:
2281:
2279:
2278:
2276:BanachâAlaoglu
2273:
2268:
2263:
2258:
2253:
2248:
2243:
2238:
2232:
2230:
2224:
2223:
2220:
2219:
2217:
2216:
2211:
2206:
2204:Locally convex
2201:
2187:
2182:
2176:
2174:
2170:
2169:
2167:
2166:
2161:
2156:
2151:
2146:
2141:
2136:
2131:
2126:
2121:
2115:
2109:
2105:
2104:
2090:
2088:
2087:
2080:
2073:
2065:
2059:
2058:
2052:
2037:
2031:
2016:
2010:
1995:
1983:(5): 722â726.
1965:
1962:
1940:
1921:
1909:
1908:
1897:
1894:
1891:
1888:
1885:
1882:
1879:
1876:
1873:
1868:
1864:
1860:
1855:
1852:
1849:
1845:
1841:
1838:
1835:
1832:
1827:
1823:
1819:
1816:
1806:
1803:
1800:
1795:
1791:
1787:
1784:
1781:
1778:
1773:
1769:
1765:
1762:
1750:), and define
1733:
1732:
1721:
1718:
1715:
1711:
1705:
1701:
1695:
1692:
1689:
1685:
1680:
1676:
1673:
1652:
1642:
1641:
1626:
1623:
1619:
1615:
1610:
1606:
1600:
1596:
1584:
1573:
1566:
1563:
1557:
1553:
1548:
1543:
1539:
1528:
1516:
1511:
1508:
1505:
1501:
1497:
1492:
1488:
1484:
1479:
1475:
1440:
1391:
1381:
1380:
1350:
1337:
1240:
1237:
1231:
1230:
1210:
1208:
1197:
1194:
1183:
1163:) â„
1138:
1137:
1126:
1123:
1118:
1115:
1112:
1109:
1106:
1102:
1098:
1095:
1048:
1045:
1042:
1039:
1036:
1031:
1027:
1021:
1018:
1015:
1011:
997:is said to be
983:
980:
972:
971:
960:
957:
954:
950:
946:
941:
938:
935:
931:
926:
922:
915:
911:
906:
902:
899:
892:
888:
883:
879:
876:
873:
853:
838:
833:
832:
819:
815:
811:
808:
803:
799:
783:
768:
759:
751:
738:
730:
719:
706:
702:restricted to
695:
688:
670:
663:
657:
656:
645:
640:
636:
632:
627:
623:
619:
615:
609:
605:
599:
596:
593:
589:
584:
580:
576:
570:
566:
560:
557:
554:
550:
545:
541:
538:
524:
523:
512:
508:
505:
502:
492:
489:
486:
483:
480:
477:
474:
471:
466:
462:
458:
453:
450:
447:
443:
439:
434:
430:
426:
421:
417:
391:
373:
372:
361:
358:
355:
350:
346:
342:
337:
333:
329:
324:
320:
316:
313:
310:
307:
304:
301:
296:
292:
288:
285:
282:
279:
276:
273:
270:
267:
264:
236:
224:
207:
206:
195:
192:
188:
184:
179:
176:
173:
169:
164:
160:
157:
102:
99:
89:discrete-time
50:, named after
29:Wold's theorem
24:
14:
13:
10:
9:
6:
4:
3:
2:
2818:
2807:
2804:
2802:
2799:
2797:
2794:
2792:
2789:
2788:
2786:
2771:
2770:
2765:
2763:
2761:
2757:
2753:
2749:
2745:
2744:
2742:
2738:
2732:
2729:
2727:
2724:
2722:
2719:
2717:
2714:
2712:
2709:
2707:
2704:
2702:
2699:
2697:
2694:
2692:
2689:
2688:
2686:
2682:
2675:
2671:
2668:
2666:
2663:
2661:
2658:
2657:
2655:
2653:Other results
2651:
2645:
2642:
2640:
2637:
2635:
2632:
2631:
2629:
2625:
2619:
2616:
2614:
2611:
2609:
2605:
2604:Hilbert space
2602:
2600:
2596:
2595:Inner product
2593:
2591:
2588:
2587:
2585:
2581:
2577:
2570:
2565:
2563:
2558:
2556:
2551:
2550:
2547:
2535:
2527:
2526:
2523:
2517:
2514:
2512:
2509:
2507:
2506:Weak topology
2504:
2502:
2499:
2497:
2494:
2492:
2489:
2488:
2486:
2482:
2475:
2471:
2468:
2466:
2463:
2461:
2458:
2456:
2453:
2451:
2448:
2446:
2443:
2441:
2438:
2436:
2433:
2431:
2430:Index theorem
2428:
2426:
2423:
2421:
2418:
2416:
2413:
2412:
2410:
2406:
2400:
2397:
2395:
2392:
2391:
2389:
2387:Open problems
2385:
2379:
2376:
2374:
2371:
2369:
2366:
2364:
2361:
2359:
2356:
2354:
2351:
2350:
2348:
2344:
2338:
2335:
2333:
2330:
2328:
2325:
2323:
2320:
2318:
2315:
2313:
2310:
2308:
2305:
2303:
2300:
2298:
2295:
2293:
2290:
2289:
2287:
2283:
2277:
2274:
2272:
2269:
2267:
2264:
2262:
2259:
2257:
2254:
2252:
2249:
2247:
2244:
2242:
2239:
2237:
2234:
2233:
2231:
2229:
2225:
2215:
2212:
2210:
2207:
2205:
2202:
2199:
2195:
2191:
2188:
2186:
2183:
2181:
2178:
2177:
2175:
2171:
2165:
2162:
2160:
2157:
2155:
2152:
2150:
2147:
2145:
2142:
2140:
2137:
2135:
2132:
2130:
2127:
2125:
2122:
2120:
2117:
2116:
2113:
2110:
2106:
2101:
2097:
2093:
2086:
2081:
2079:
2074:
2072:
2067:
2066:
2063:
2055:
2053:0-19-503591-7
2049:
2045:
2044:
2038:
2034:
2032:0-12-221350-5
2028:
2024:
2023:
2017:
2013:
2011:3-7643-5285-X
2007:
2003:
2002:
1996:
1991:
1986:
1982:
1979:
1978:
1973:
1968:
1967:
1963:
1961:
1959:
1955:
1951:
1947:
1943:
1936:
1932:
1928:
1924:
1916:
1914:
1895:
1889:
1883:
1880:
1874:
1871:
1866:
1862:
1853:
1850:
1847:
1843:
1839:
1833:
1830:
1825:
1821:
1801:
1793:
1789:
1779:
1771:
1767:
1763:
1753:
1752:
1751:
1749:
1745:
1741:
1738:
1719:
1716:
1713:
1709:
1703:
1699:
1693:
1690:
1687:
1683:
1678:
1674:
1671:
1664:
1663:
1662:
1660:
1655:
1651:
1647:
1624:
1621:
1617:
1613:
1608:
1604:
1598:
1594:
1585:
1571:
1561:
1555:
1551:
1546:
1541:
1537:
1529:
1514:
1509:
1506:
1503:
1499:
1495:
1490:
1486:
1482:
1477:
1473:
1465:
1464:
1463:
1460:
1458:
1454:
1450:
1445:
1443:
1436:
1432:
1428:
1425:
1421:
1419:
1415:
1411:
1407:
1403:
1399:
1394:
1390:
1386:
1378:
1374:
1370:
1366:
1362:
1358:
1353:
1349:
1345:
1340:
1336:
1332:
1328:
1325:
1324:
1323:
1321:
1317:
1313:
1309:
1305:
1301:
1296:
1294:
1290:
1286:
1282:
1278:
1274:
1270:
1267:generated by
1266:
1262:
1258:
1255:). Denote by
1254:
1250:
1246:
1238:
1236:
1227:
1218:
1214:
1211:This section
1209:
1206:
1202:
1201:
1195:
1193:
1191:
1186:
1182:
1178:
1175: â
1174:
1170:
1166:
1162:
1158:
1154:
1150:
1146:
1141:
1124:
1121:
1116:
1113:
1110:
1107:
1104:
1100:
1096:
1093:
1086:
1085:
1084:
1082:
1078:
1074:
1070:
1066:
1062:
1046:
1040:
1034:
1029:
1025:
1019:
1016:
1013:
1009:
1000:
996:
991:
989:
981:
979:
977:
958:
955:
952:
948:
944:
939:
936:
933:
929:
924:
920:
913:
909:
900:
897:
890:
886:
877:
874:
871:
864:
863:
862:
860:
856:
849:
845:
841:
817:
813:
809:
806:
801:
797:
789:
788:
787:
782:
778:
775:
771:
762:
758:
754:
747:
741:
737:
733:
726:
722:
714:
712:
705:
701:
694:
687:
683:
678:
676:
669:
662:
643:
638:
634:
630:
625:
621:
617:
613:
607:
603:
597:
594:
591:
587:
582:
578:
574:
568:
564:
558:
555:
552:
548:
543:
539:
536:
529:
528:
527:
510:
506:
503:
500:
484:
478:
475:
472:
464:
460:
456:
451:
448:
445:
441:
437:
432:
428:
424:
419:
415:
407:
406:
405:
403:
399:
396: =
394:
390:
386:
382:
378:
359:
356:
353:
348:
344:
340:
335:
331:
327:
322:
318:
314:
311:
308:
302:
294:
290:
286:
280:
274:
271:
268:
265:
262:
255:
254:
253:
251:
247:
242:
239:
235:
231:
227:
220:
216:
212:
193:
190:
186:
182:
177:
174:
171:
167:
162:
158:
155:
148:
147:
146:
144:
140:
136:
132:
128:
124:
120:
116:
112:
111:Hilbert space
108:
100:
98:
96:
92:
88:
84:
79:
77:
73:
69:
68:Hilbert space
65:
61:
57:
53:
49:
45:
41:
37:
30:
19:
2768:
2759:
2755:
2751:
2747:
2716:Self-adjoint
2627:Main results
2496:Balanced set
2470:Distribution
2408:Applications
2261:KreinâMilman
2246:Closed graph
2042:
2021:
2000:
1980:
1975:
1957:
1953:
1949:
1948:= 0 implies
1945:
1938:
1934:
1930:
1926:
1919:
1917:
1912:
1910:
1747:
1743:
1739:
1734:
1658:
1653:
1649:
1645:
1643:
1461:
1456:
1452:
1448:
1446:
1438:
1434:
1430:
1426:
1423:
1422:
1413:
1409:
1405:
1401:
1397:
1392:
1388:
1384:
1382:
1372:
1368:
1364:
1360:
1351:
1347:
1343:
1338:
1334:
1330:
1326:
1319:
1315:
1311:
1307:
1303:
1299:
1297:
1292:
1288:
1284:
1280:
1276:
1272:
1268:
1260:
1256:
1252:
1248:
1244:
1242:
1234:
1221:
1217:adding to it
1212:
1189:
1184:
1180:
1176:
1172:
1168:
1164:
1160:
1156:
1152:
1147:is called a
1144:
1142:
1139:
1080:
1076:
1072:
1068:
1064:
1061:multiplicity
1060:
998:
994:
993:An isometry
992:
985:
975:
973:
861:. Therefore
858:
851:
847:
843:
836:
834:
780:
776:
766:
760:
756:
749:
745:
739:
735:
728:
724:
717:
715:
710:
703:
699:
692:
685:
681:
679:
674:
667:
660:
658:
525:
401:
397:
392:
388:
384:
380:
376:
374:
249:
245:
243:
237:
233:
229:
222:
214:
210:
208:
142:
138:
134:
130:
126:
122:
118:
114:
106:
104:
80:
47:
43:
33:
2801:C*-algebras
2726:Trace class
2425:Heat kernel
2415:Hardy space
2322:Trace class
2236:HahnâBanach
2198:Topological
1937:) and thus
1640:is compact.
1143:A subspace
835:where each
66:on a given
52:Herman Wold
36:mathematics
2785:Categories
2358:C*-algebra
2173:Properties
1964:References
1265:C*-algebra
1171:) for all
87:stationary
2332:Unbounded
2327:Transpose
2285:Operators
2214:Separable
2209:Reflexive
2194:Algebraic
2180:Barrelled
1881:⊕
1851:∈
1848:α
1844:⨁
1815:Φ
1794:∗
1786:→
1772:∗
1761:Φ
1714:⊕
1691:∈
1688:α
1684:⨁
1614:−
1565:¯
1547:∗
1224:June 2008
1114:≤
1111:α
1108:≤
1101:⨁
1017:≥
1010:⋂
953:⊕
937:∈
934:α
930:⨁
898:⊕
818:α
810:⊕
748:"shifts"
631:⊕
595:≥
588:⋂
579:⊕
556:≥
549:⨁
504:≥
476:⊖
438:⊖
357:⋯
354:⊃
341:⊃
328:⊃
312:⋯
309:⊃
287:⊃
272:⊃
191:⊕
175:∈
172:α
168:⨁
2740:Examples
2534:Category
2346:Algebras
2228:Theorems
2185:Complete
2154:Schwartz
2100:glossary
1363:∈
988:spectrum
772:is some
213:, where
2754:) with
2731:Unitary
2590:Adjoint
2337:Unitary
2317:Nuclear
2302:Compact
2297:Bounded
2292:Adjoint
2266:Minâmax
2159:Sobolev
2144:Nuclear
2134:Hilbert
2129:Fréchet
2094: (
1271:, i.e.
982:Remarks
217:is the
101:Details
58:, is a
2711:Normal
2312:Normal
2149:Orlicz
2139:Hölder
2119:Banach
2108:Spaces
2096:topics
2050:
2029:
2008:
1396:where
1371:) and
1263:) the
375:where
228:, and
125:, and
74:and a
2762:<â
2124:Besov
1375:is a
1355:is a
1333:) = {
526:then
109:be a
2684:Maps
2606:and
2597:and
2472:(or
2190:Dual
2048:ISBN
2027:ISBN
2006:ISBN
1298:Let
1283:and
1059:The
999:pure
846:and
734:and
691:) =
666:and
105:Let
62:for
54:and
1985:doi
1295:).
1219:.
1155:if
1151:of
755:to
680:So
496:for
81:In
46:or
34:In
2787::
2098:â
1981:73
1974:.
1954:C*
1944:+
1929:â
1915::
1810:by
1742:â
1661::
1459:.
1444:.
1427:C*
1420:.
1410:C*
1387:=
1379:}.
1346:|
1342:+
1327:C*
1312:C*
1289:C*
1285:V*
1273:C*
1257:C*
1247:â
1192:.
1069:V*
978:.
763:+1
744::
742:+1
713:.
677:.
129:â
113:,
97:.
78:.
42:,
2769:F
2760:n
2756:K
2752:K
2750:(
2748:C
2676:)
2672:(
2568:e
2561:t
2554:v
2476:)
2200:)
2196:/
2192:(
2102:)
2084:e
2077:t
2070:v
2056:.
2035:.
2014:.
1993:.
1987::
1958:V
1956:(
1950:f
1946:K
1941:f
1939:T
1935:T
1933:(
1931:C
1927:f
1922:f
1920:T
1913:V
1896:.
1893:)
1890:U
1887:(
1884:f
1878:)
1875:K
1872:+
1867:f
1863:T
1859:(
1854:A
1840:=
1837:)
1834:K
1831:+
1826:f
1822:T
1818:(
1805:)
1802:V
1799:(
1790:C
1783:)
1780:S
1777:(
1768:C
1764::
1748:U
1746:(
1744:f
1740:f
1720:.
1717:U
1710:)
1704:z
1700:T
1694:A
1679:(
1675:=
1672:V
1659:U
1654:z
1650:T
1646:V
1625:g
1622:f
1618:T
1609:g
1605:T
1599:f
1595:T
1572:.
1562:f
1556:T
1552:=
1542:f
1538:T
1515:.
1510:g
1507:+
1504:f
1500:T
1496:=
1491:g
1487:T
1483:+
1478:f
1474:T
1457:T
1453:T
1451:(
1449:C
1441:z
1439:T
1435:V
1431:V
1429:(
1414:S
1412:(
1406:T
1404:(
1402:C
1398:z
1393:z
1389:T
1385:S
1373:K
1369:T
1367:(
1365:C
1361:f
1352:f
1348:T
1344:K
1339:f
1335:T
1331:S
1329:(
1320:S
1316:S
1314:(
1308:T
1304:T
1302:(
1300:C
1293:V
1291:(
1281:V
1277:V
1275:(
1269:V
1261:V
1259:(
1253:H
1251:(
1249:L
1245:V
1226:)
1222:(
1190:V
1185:i
1181:M
1177:m
1173:n
1169:M
1167:(
1165:V
1161:M
1159:(
1157:V
1153:V
1145:M
1125:.
1122:S
1117:N
1105:1
1097:=
1094:V
1081:N
1077:V
1073:A
1065:V
1047:.
1044:}
1041:0
1038:{
1035:=
1030:i
1026:H
1020:0
1014:i
995:V
976:V
959:,
956:U
949:)
945:S
940:A
925:(
921:=
914:2
910:K
905:|
901:V
891:1
887:K
882:|
878:V
875:=
872:V
859:S
854:α
852:H
848:V
844:V
839:α
837:H
814:H
807:=
802:1
798:K
784:1
781:K
777:α
769:i
767:M
761:i
757:M
752:i
750:M
746:V
740:i
736:M
731:i
729:M
725:V
720:i
718:M
711:U
707:2
704:K
700:V
696:2
693:K
689:2
686:K
684:(
682:V
675:V
671:2
668:K
664:1
661:K
644:.
639:2
635:K
626:1
622:K
618:=
614:)
608:i
604:H
598:0
592:i
583:(
575:)
569:i
565:M
559:0
553:i
544:(
540:=
537:H
511:,
507:0
501:i
491:)
488:)
485:H
482:(
479:V
473:H
470:(
465:i
461:V
457:=
452:1
449:+
446:i
442:H
433:i
429:H
425:=
420:i
416:M
402:H
400:(
398:V
393:i
389:H
385:V
381:H
379:(
377:V
360:,
349:2
345:H
336:1
332:H
323:0
319:H
315:=
306:)
303:H
300:(
295:2
291:V
284:)
281:H
278:(
275:V
269:H
266:=
263:H
250:H
246:V
238:α
234:H
230:U
225:α
223:H
215:S
211:A
194:U
187:)
183:S
178:A
163:(
159:=
156:V
143:V
135:H
133:(
131:L
127:V
123:H
119:H
117:(
115:L
107:H
31:.
20:)
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