2542:
1821:
2537:{\displaystyle {\begin{aligned}\operatorname {tr} (XX^{*})&=\operatorname {tr} \left(P_{V}TT^{*}P_{V}-P_{V}TP_{V}T^{*}P_{V}\right)\\&=\operatorname {tr} (P_{V}TT^{*}P_{V})-\operatorname {tr} (P_{V}TP_{V}T^{*}P_{V})\\&=\operatorname {tr} (P_{V}^{2}TT^{*})-\operatorname {tr} (P_{V}^{2}TP_{V}T^{*})\\&=\operatorname {tr} (P_{V}TT^{*})-\operatorname {tr} (P_{V}TP_{V}T^{*})\\&=\operatorname {tr} (P_{V}TT^{*})-\operatorname {tr} (TP_{V}T^{*})&&{\text{using the hypothesis that }}T{\text{ stabilizes }}V\\&=\operatorname {tr} (P_{V}TT^{*})-\operatorname {tr} (P_{V}T^{*}T)\\&=\operatorname {tr} (P_{V}(TT^{*}-T^{*}T))\\&=0.\end{aligned}}}
4490:
25:
1806:
1112:
Eigenvectors of a normal operator corresponding to different eigenvalues are orthogonal, and a normal operator stabilizes the orthogonal complement of each of its eigenspaces. This implies the usual spectral theorem: every normal operator on a finite-dimensional space is diagonalizable by a unitary
1529:
2555:) suitably interpreted. However, for bounded normal operators, the orthogonal complement to a stable subspace may not be stable. It follows that the Hilbert space cannot in general be spanned by eigenvectors of a normal operator. Consider, for example, the
561:
2846:
The success of the theory of normal operators led to several attempts for generalization by weakening the commutativity requirement. Classes of operators that include normal operators are (in order of inclusion)
1801:{\displaystyle {\begin{aligned}XX^{*}&=P_{V}T({\boldsymbol {1}}_{H}-P_{V})^{2}T^{*}P_{V}\\&=P_{V}T({\boldsymbol {1}}_{H}-P_{V})T^{*}P_{V}\\&=P_{V}TT^{*}P_{V}-P_{V}TP_{V}T^{*}P_{V}.\end{aligned}}}
2833:
1826:
1534:
770:
708:
2765:
1080:
833:
1305:
414:
2688:
1249:
653:
361:
2584:
1033:
285:
1110:
1180:
1153:
970:
903:
439:
3641:
1013:
1200:
1053:
990:
943:
923:
876:
856:
604:
584:
434:
332:
309:
3108:
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945:
is injective. Put in another way, the kernel of a normal operator is the orthogonal complement of its range. It follows that the kernel of the operator
4379:
3743:
2995:
3376:
4215:
4042:
3398:
3061:
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4205:
3381:
3154:
3403:
4332:
4187:
3913:
4163:
3806:
3728:
3391:
3621:
2838:
The spectral theorem still holds for unbounded (normal) operators. The proofs work by reduction to bounded (normal) operators.
2776:
2547:
The same argument goes through for compact normal operators in infinite dimensional
Hilbert spaces, where one make use of the
89:
3474:
3272:
2989:
2634:
The definition of normal operators naturally generalizes to some class of unbounded operators. Explicitly, a closed operator
61:
3469:
2909:
4055:
68:
4144:
4035:
3930:
3626:
713:
274:
108:
4414:
3444:
42:
4059:
3413:
658:
75:
3327:
3211:
144:
46:
2718:
4210:
3878:
3636:
3147:
2888:
2548:
1120:
The product of normal operators that commute is again normal; this is nontrivial, but follows directly from
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775:
57:
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4266:
4200:
4028:
3945:
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3262:
1254:
2979:
4230:
3799:
3773:
3693:
3247:
4475:
4429:
4353:
4235:
3986:
3899:
3748:
3646:
3526:
2883:
368:
1113:
operator. There is also an infinite-dimensional version of the spectral theorem expressed in terms of
4514:
4470:
4286:
3753:
3616:
3449:
3434:
3242:
3206:
1114:
3940:
2644:
1205:
609:
4519:
4322:
4220:
4123:
4006:
3935:
3345:
3335:
3216:
3140:
278:
187:
holds for them. The class of normal operators is well understood. Examples of normal operators are
4419:
4195:
3708:
3683:
3501:
3490:
3201:
35:
339:
4450:
4394:
4358:
3904:
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3792:
3559:
3549:
3544:
3252:
3054:
2562:
1812:
1018:
4157:
3955:
3909:
3852:
3304:
3018:
2590:
556:{\displaystyle \|Tx\|^{2}=\langle T^{*}Tx,x\rangle =\langle TT^{*}x,x\rangle =\|T^{*}x\|^{2}}
82:
4153:
1085:
4433:
3838:
3718:
3697:
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2913:
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1158:
1131:
1121:
948:
881:
4020:
8:
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4337:
4051:
3834:
3521:
3257:
2861:
2851:
281:
126:
995:
4424:
4291:
3651:
3580:
3511:
3355:
3317:
3035:
2871:
1185:
1038:
975:
928:
908:
861:
841:
589:
569:
419:
317:
294:
204:
4404:
3960:
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3829:
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3733:
3418:
3340:
3039:
2985:
1323:
230:
166:
4409:
4327:
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4261:
4256:
4251:
3970:
3847:
3763:
3464:
3312:
3267:
3191:
3027:
3016:
AndĂŽ, Tsuyoshi (1963). "Note on invariant subspaces of a compact normal operator".
1312:
270:
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184:
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4139:
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1316:
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3585:
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3186:
3123:
217:
134:
2589:
The invariant subspaces of a shift acting on Hardy space are characterized by
4508:
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4389:
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1512:
254:
137:
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3105:
3094:
2945:(2nd ed.), Englewood Cliffs, N.J.: Prentice-Hall, Inc., p. 312,
4384:
4374:
4281:
4083:
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3703:
3288:
122:
3126:, A Course in Functional Analysis, Second Edition, Chapter X, Section §4
4317:
4149:
3196:
3031:
2623:
162:
3181:
3167:
2938:
2701:
be dense, and the equality includes the assertion that the domain of
2601:
The notion of normal operators generalizes to an involutive algebra:
905:
have the same kernel and the same range. Consequently, the range of
24:
3768:
3713:
3084:
Weidmann, Lineare
Operatoren in HilbertrÀumen, Chapter 4, Section 3
1015:
Every generalized eigenvalue of a normal operator is thus genuine.
257:
is the matrix expression of a normal operator on the
Hilbert space
3784:
3093:
Alexander Frei, Spectral
Measures, Mathematics Stack Exchange,
2977:
2712:
Equivalently normal operators are precisely those for which
4050:
2956:
2828:{\displaystyle {\mathcal {D}}(N)={\mathcal {D}}(N^{*}).}
1329:
1117:. The residual spectrum of a normal operator is empty.
16:(on a complex Hilbert space) continuous linear operator
2779:
2721:
2647:
2622:
The most important case is when such an algebra is a
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1824:
1532:
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1208:
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1134:
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1021:
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978:
951:
931:
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864:
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778:
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311:
be a bounded operator. The following are equivalent.
297:
1350:, then it also stabilizes its orthogonal complement
1342:
real or complex
Hilbert space (inner product space)
49:. Unsourced material may be challenged and removed.
4380:Spectral theory of ordinary differential equations
3744:Spectral theory of ordinary differential equations
3162:
2981:Linear Operator Theory in Engineering and Sciences
2827:
2759:
2682:
2578:
2536:
1800:
1311:The operator norm of a normal operator equals its
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1243:
1194:
1174:
1147:
1124:, which states (in a form generalized by Putnam):
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1007:
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827:
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647:
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555:
428:
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355:
326:
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3642:SchröderâBernstein theorems for operator algebras
2608:of an involutive algebra is said to be normal if
4506:
2709:, which is not necessarily the case in general.
566:The self-adjoint and antiâself adjoint parts of
1354:. (This statement is trivial in the case where
765:{\displaystyle i\,T_{2}:={\frac {T-T^{*}}{2}},}
3015:
2619:Self-adjoint and unitary elements are normal.
2596:
4036:
3800:
3148:
3052:
2629:
2962:
2936:
2891: â Bounded operators with sub-unit norm
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2586:, which is normal, but has no eigenvalues.
703:{\displaystyle T_{1}:={\frac {T+T^{*}}{2}}}
183:Normal operators are important because the
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4029:
3807:
3793:
3155:
3141:
269:Normal operators are characterized by the
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109:Learn how and when to remove this message
4333:Group algebra of a locally compact group
2978:Naylor, Arch W.; Sell George R. (1982).
2973:
2971:
2908:In contrast, for the important class of
2760:{\displaystyle \|Nx\|=\|N^{*}x\|\qquad }
1815:and of orthogonal projections we have:
1202:is a bounded linear operator such that
277:(in particular, a normal operator on a
4507:
3046:
1375:. Then the orthogonal projection onto
1075:{\displaystyle {\overline {\lambda }}}
1035:is an eigenvalue of a normal operator
828:{\displaystyle T_{1}T_{2}=T_{2}T_{1}.}
4024:
3788:
3475:Spectral theory of normal C*-algebras
3273:Spectral theory of normal C*-algebras
3136:
3117:
2968:
1330:Properties in finite-dimensional case
1322:A normal operator coincides with its
1300:{\displaystyle N_{1}^{*}A=AN_{2}^{*}}
1055:if and only if its complex conjugate
3470:Spectral theory of compact operators
47:adding citations to reliable sources
18:
3087:
2910:Creation and annihilation operators
2693:Here, the existence of the adjoint
13:
3814:
3622:CohenâHewitt factorization theorem
2801:
2782:
1371:be the orthogonal projection onto
14:
4531:
3931:Compact operator on Hilbert space
3627:Extensions of symmetric operators
2841:
1515:on the space of endomorphisms of
409:{\displaystyle \|Tx\|=\|T^{*}x\|}
4489:
4488:
4415:Topological quantum field theory
3445:Positive operator-valued measure
1656:
1575:
207:(i.e., self-adjoint operators):
23:
3729:RayleighâFaberâKrahn inequality
3067:from the original on 2011-09-18
2998:from the original on 2021-06-26
2756:
2559:(or two-sided shift) acting on
2353:using the hypothesis that
1519:, it is enough to show that tr(
34:needs additional citations for
3078:
3009:
2930:
2902:
2819:
2806:
2793:
2787:
2683:{\displaystyle N^{*}N=NN^{*}.}
2514:
2511:
2479:
2466:
2447:
2421:
2409:
2383:
2345:
2319:
2307:
2281:
2262:
2226:
2214:
2188:
2169:
2128:
2116:
2085:
2066:
2020:
2008:
1972:
1851:
1835:
1679:
1651:
1599:
1570:
1244:{\displaystyle N_{1}A=AN_{2},}
648:{\displaystyle T=T_{1}+iT_{2}}
1:
4211:Uniform boundedness principle
3637:Limiting absorption principle
3055:"Operators on Hilbert spaces"
2923:
2889:Contraction (operator theory)
2549:Hilbert-Schmidt inner product
264:
3263:Singular value decomposition
2697:requires that the domain of
1811:Now using properties of the
1182:are normal operators and if
1067:
7:
3694:Hearing the shape of a drum
3377:Decomposition of a spectrum
2877:
2597:Normal elements of algebras
1444:. The goal is to show that
858:is a normal operator, then
10:
4536:
4354:Invariant subspace problem
3900:Hilbert projection theorem
3282:Special Elements/Operators
2963:Hoffman & Kunze (1971)
2884:Continuous linear operator
2630:Unbounded normal operators
1523:) = 0. First we note that
1115:projection-valued measures
356:{\displaystyle T^{\star }}
4484:
4443:
4367:
4346:
4305:
4244:
4186:
4132:
4074:
4067:
3979:
3923:
3892:
3879:CauchyâSchwarz inequality
3866:
3822:
3754:Superstrong approximation
3676:
3660:
3617:Banach algebra cohomology
3604:
3568:
3537:
3483:
3450:Projection-valued measure
3435:Borel functional calculus
3427:
3369:
3326:
3281:
3235:
3207:Projection-valued measure
3174:
2638:is said to be normal if
2579:{\displaystyle \ell ^{2}}
4323:Spectrum of a C*-algebra
3346:Spectrum of a C*-algebra
3217:Spectrum of a C*-algebra
2895:
1028:{\displaystyle \lambda }
925:is dense if and only if
286:unitarily diagonalizable
4420:Noncommutative geometry
3774:WienerâKhinchin theorem
3709:Kuznetsov trace formula
3684:Almost Mathieu operator
3502:Banach function algebra
3491:Amenable Banach algebra
3248:GelfandâNaimark theorem
3202:Noncommutative topology
972:coincides with that of
275:compact normal operator
4476:TomitaâTakesaki theory
4451:Approximation property
4395:Calculus of variations
3749:SturmâLiouville theory
3647:ShermanâTakeda theorem
3527:TomitaâTakesaki theory
3302:Hermitian/Self-adjoint
3253:Gelfand representation
3053:Garrett, Paul (2005).
2984:. New York: Springer.
2829:
2761:
2684:
2580:
2538:
2361: stabilizes
1802:
1346:stabilizes a subspace
1301:
1245:
1196:
1176:
1149:
1106:
1105:{\displaystyle N^{*}.}
1076:
1049:
1029:
1009:
986:
966:
939:
919:
899:
872:
852:
829:
766:
704:
649:
600:
580:
557:
430:
410:
357:
328:
305:
4471:BanachâMazur distance
4434:Generalized functions
3910:Polarization identity
3853:Orthogonal complement
3243:GelfandâMazur theorem
3019:Archiv der Mathematik
2867:Quasinormal operators
2830:
2762:
2685:
2581:
2539:
1803:
1403:can be expressed as (
1334:If a normal operator
1302:
1246:
1197:
1177:
1175:{\displaystyle N_{2}}
1150:
1148:{\displaystyle N_{1}}
1107:
1077:
1050:
1030:
1010:
987:
967:
965:{\displaystyle N^{k}}
940:
920:
900:
898:{\displaystyle N^{*}}
873:
853:
830:
767:
705:
650:
601:
586:commute. That is, if
581:
558:
431:
411:
358:
329:
306:
4216:Kakutani fixed-point
4201:Riesz representation
3884:Riesz representation
3839:L-semi-inner product
3719:Proto-value function
3698:Dirichlet eigenvalue
3612:Abstract index group
3497:Approximate identity
3460:Rigged Hilbert space
3336:KreinâRutman theorem
3182:Involution/*-algebra
2916:, they don't commute
2914:quantum field theory
2862:Paranormal operators
2852:Hyponormal operators
2777:
2719:
2645:
2563:
1822:
1530:
1255:
1206:
1186:
1159:
1132:
1086:
1082:is an eigenvalue of
1059:
1039:
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996:
976:
949:
929:
909:
882:
862:
842:
776:
714:
659:
610:
590:
570:
440:
420:
369:
340:
318:
295:
43:improve this article
4400:Functional calculus
4359:Mahler's conjecture
4338:Von Neumann algebra
4052:Functional analysis
3905:Parseval's identity
3874:Bessel's inequality
3522:Von Neumann algebra
3258:Polar decomposition
2872:Subnormal operators
2145:
2102:
1296:
1272:
282:inner product space
205:Hermitian operators
127:functional analysis
4425:Riemann hypothesis
4124:Topological vector
3652:Unbounded operator
3581:Essential spectrum
3560:SchurâHorn theorem
3550:BauerâFike theorem
3545:AlonâBoppana bound
3538:Finite-Dimensional
3512:Nuclear C*-algebra
3356:Spectral asymmetry
3111:2021-06-26 at the
3100:2021-06-26 at the
3032:10.1007/BF01234964
2937:Hoffman, Kenneth;
2825:
2757:
2680:
2591:Beurling's theorem
2576:
2534:
2532:
2131:
2088:
1798:
1796:
1358:is self-adjoint.)
1340:finite-dimensional
1297:
1282:
1258:
1241:
1192:
1172:
1145:
1102:
1072:
1045:
1025:
1008:{\displaystyle k.}
1005:
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962:
935:
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868:
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825:
762:
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645:
596:
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553:
426:
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324:
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279:finite-dimensional
231:positive operators
4502:
4501:
4405:Integral operator
4182:
4181:
4018:
4017:
3961:Sesquilinear form
3914:Parallelogram law
3858:Orthonormal basis
3782:
3781:
3759:Transfer operator
3734:Spectral geometry
3419:Spectral abscissa
3399:Approximate point
3341:Normal eigenvalue
2991:978-0-387-95001-3
2362:
2354:
1324:Aluthge transform
1195:{\displaystyle A}
1122:Fuglede's theorem
1070:
1048:{\displaystyle N}
985:{\displaystyle N}
938:{\displaystyle N}
918:{\displaystyle N}
871:{\displaystyle N}
851:{\displaystyle N}
757:
698:
599:{\displaystyle T}
579:{\displaystyle T}
429:{\displaystyle x}
327:{\displaystyle T}
304:{\displaystyle T}
249:is self-adjoint).
192:unitary operators
167:Hermitian adjoint
119:
118:
111:
93:
58:"Normal operator"
4527:
4515:Linear operators
4492:
4491:
4410:Jones polynomial
4328:Operator algebra
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4038:
4031:
4022:
4021:
3848:Prehilbert space
3809:
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3795:
3786:
3785:
3764:Transform theory
3484:Special algebras
3465:Spectral theorem
3428:Spectral Theorem
3268:Spectral theorem
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2585:
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2582:
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2575:
2574:
2551:, defined by tr(
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2535:
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2520:
2507:
2506:
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2407:
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2250:
2238:
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2213:
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2199:
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2168:
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2157:
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2139:
2115:
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1607:
1606:
1597:
1596:
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1583:
1578:
1566:
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1549:
1548:
1395:. The fact that
1313:numerical radius
1306:
1304:
1303:
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1290:
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1266:
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1052:
1051:
1046:
1034:
1032:
1031:
1026:
1014:
1012:
1011:
1006:
991:
989:
988:
983:
971:
969:
968:
963:
961:
960:
944:
942:
941:
936:
924:
922:
921:
916:
904:
902:
901:
896:
894:
893:
877:
875:
874:
869:
857:
855:
854:
849:
834:
832:
831:
826:
821:
820:
811:
810:
798:
797:
788:
787:
771:
769:
768:
763:
758:
753:
752:
751:
735:
730:
729:
709:
707:
706:
701:
699:
694:
693:
692:
676:
671:
670:
654:
652:
651:
646:
644:
643:
628:
627:
605:
603:
602:
597:
585:
583:
582:
577:
562:
560:
559:
554:
552:
551:
539:
538:
511:
510:
477:
476:
461:
460:
435:
433:
432:
427:
415:
413:
412:
407:
399:
398:
362:
360:
359:
354:
352:
351:
333:
331:
330:
325:
310:
308:
307:
302:
271:spectral theorem
185:spectral theorem
114:
107:
103:
100:
94:
92:
51:
27:
19:
4535:
4534:
4530:
4529:
4528:
4526:
4525:
4524:
4520:Operator theory
4505:
4504:
4503:
4498:
4480:
4444:Advanced topics
4439:
4363:
4342:
4301:
4267:HilbertâSchmidt
4240:
4231:GelfandâNaimark
4178:
4128:
4063:
4049:
4019:
4014:
4007:SegalâBargmann
3975:
3946:HilbertâSchmidt
3936:Densely defined
3919:
3888:
3862:
3818:
3813:
3783:
3778:
3739:Spectral method
3724:Ramanujan graph
3672:
3656:
3632:Fredholm theory
3600:
3595:Shilov boundary
3591:Structure space
3569:Generalizations
3564:
3555:Numerical range
3533:
3517:Uniform algebra
3479:
3455:Riesz projector
3440:Min-max theorem
3423:
3409:Direct integral
3365:
3351:Spectral radius
3322:
3277:
3231:
3222:Spectral radius
3170:
3164:Spectral theory
3161:
3131:
3130:
3122:
3118:
3113:Wayback Machine
3102:Wayback Machine
3092:
3088:
3083:
3079:
3070:
3068:
3064:
3057:
3051:
3047:
3014:
3010:
3001:
2999:
2992:
2976:
2969:
2961:
2957:
2935:
2931:
2926:
2921:
2920:
2907:
2903:
2898:
2880:
2844:
2813:
2809:
2800:
2799:
2781:
2780:
2778:
2775:
2774:
2744:
2740:
2720:
2717:
2716:
2705:equals that of
2671:
2667:
2652:
2648:
2646:
2643:
2642:
2632:
2599:
2570:
2566:
2564:
2561:
2560:
2557:bilateral shift
2531:
2530:
2518:
2517:
2502:
2498:
2489:
2485:
2473:
2469:
2451:
2450:
2438:
2434:
2428:
2424:
2403:
2399:
2390:
2386:
2368:
2367:
2359:
2351:
2348:
2339:
2335:
2329:
2325:
2301:
2297:
2288:
2284:
2266:
2265:
2256:
2252:
2246:
2242:
2233:
2229:
2208:
2204:
2195:
2191:
2173:
2172:
2163:
2159:
2153:
2149:
2140:
2135:
2110:
2106:
2097:
2092:
2070:
2069:
2060:
2056:
2050:
2046:
2040:
2036:
2027:
2023:
2002:
1998:
1992:
1988:
1979:
1975:
1957:
1956:
1945:
1941:
1935:
1931:
1925:
1921:
1912:
1908:
1899:
1895:
1889:
1885:
1876:
1872:
1871:
1867:
1854:
1845:
1841:
1825:
1823:
1820:
1819:
1795:
1794:
1785:
1781:
1775:
1771:
1765:
1761:
1752:
1748:
1739:
1735:
1729:
1725:
1716:
1712:
1703:
1702:
1696:
1692:
1686:
1682:
1673:
1669:
1660:
1655:
1654:
1642:
1638:
1629:
1628:
1622:
1618:
1612:
1608:
1602:
1598:
1592:
1588:
1579:
1574:
1573:
1561:
1557:
1550:
1544:
1540:
1533:
1531:
1528:
1527:
1497:
1491:
1480:
1466:
1460:
1449:
1442:
1438:
1431:
1424:
1417:
1411:
1393:
1387:
1369:
1332:
1317:spectral radius
1291:
1286:
1267:
1262:
1256:
1253:
1252:
1232:
1228:
1213:
1209:
1207:
1204:
1203:
1187:
1184:
1183:
1166:
1162:
1160:
1157:
1156:
1139:
1135:
1133:
1130:
1129:
1093:
1089:
1087:
1084:
1083:
1062:
1060:
1057:
1056:
1040:
1037:
1036:
1020:
1017:
1016:
997:
994:
993:
977:
974:
973:
956:
952:
950:
947:
946:
930:
927:
926:
910:
907:
906:
889:
885:
883:
880:
879:
863:
860:
859:
843:
840:
839:
816:
812:
806:
802:
793:
789:
783:
779:
777:
774:
773:
747:
743:
736:
734:
725:
721:
715:
712:
711:
688:
684:
677:
675:
666:
662:
660:
657:
656:
639:
635:
623:
619:
611:
608:
607:
591:
588:
587:
571:
568:
567:
547:
543:
534:
530:
506:
502:
472:
468:
456:
452:
441:
438:
437:
421:
418:
417:
394:
390:
370:
367:
366:
347:
343:
341:
338:
337:
319:
316:
315:
296:
293:
292:
267:
148:linear operator
131:normal operator
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
4533:
4523:
4522:
4517:
4500:
4499:
4497:
4496:
4485:
4482:
4481:
4479:
4478:
4473:
4468:
4463:
4461:Choquet theory
4458:
4453:
4447:
4445:
4441:
4440:
4438:
4437:
4427:
4422:
4417:
4412:
4407:
4402:
4397:
4392:
4387:
4382:
4377:
4371:
4369:
4365:
4364:
4362:
4361:
4356:
4350:
4348:
4344:
4343:
4341:
4340:
4335:
4330:
4325:
4320:
4315:
4313:Banach algebra
4309:
4307:
4303:
4302:
4300:
4299:
4294:
4289:
4284:
4279:
4274:
4269:
4264:
4259:
4254:
4248:
4246:
4242:
4241:
4239:
4238:
4236:BanachâAlaoglu
4233:
4228:
4223:
4218:
4213:
4208:
4203:
4198:
4192:
4190:
4184:
4183:
4180:
4179:
4177:
4176:
4171:
4166:
4164:Locally convex
4161:
4147:
4142:
4136:
4134:
4130:
4129:
4127:
4126:
4121:
4116:
4111:
4106:
4101:
4096:
4091:
4086:
4081:
4075:
4069:
4065:
4064:
4048:
4047:
4040:
4033:
4025:
4016:
4015:
4013:
4012:
4004:
3998:compact &
3983:
3981:
3977:
3976:
3974:
3973:
3968:
3963:
3958:
3953:
3948:
3943:
3941:Hermitian form
3938:
3933:
3927:
3925:
3921:
3920:
3918:
3917:
3907:
3902:
3896:
3894:
3890:
3889:
3887:
3886:
3881:
3876:
3870:
3868:
3864:
3863:
3861:
3860:
3855:
3850:
3841:
3832:
3826:
3824:
3823:Basic concepts
3820:
3819:
3816:Hilbert spaces
3812:
3811:
3804:
3797:
3789:
3780:
3779:
3777:
3776:
3771:
3766:
3761:
3756:
3751:
3746:
3741:
3736:
3731:
3726:
3721:
3716:
3711:
3706:
3701:
3691:
3689:Corona theorem
3686:
3680:
3678:
3674:
3673:
3671:
3670:
3668:Wiener algebra
3664:
3662:
3658:
3657:
3655:
3654:
3649:
3644:
3639:
3634:
3629:
3624:
3619:
3614:
3608:
3606:
3602:
3601:
3599:
3598:
3588:
3586:Pseudospectrum
3583:
3578:
3576:Dirac spectrum
3572:
3570:
3566:
3565:
3563:
3562:
3557:
3552:
3547:
3541:
3539:
3535:
3534:
3532:
3531:
3530:
3529:
3519:
3514:
3509:
3504:
3499:
3493:
3487:
3485:
3481:
3480:
3478:
3477:
3472:
3467:
3462:
3457:
3452:
3447:
3442:
3437:
3431:
3429:
3425:
3424:
3422:
3421:
3416:
3411:
3406:
3401:
3396:
3395:
3394:
3389:
3384:
3373:
3371:
3367:
3366:
3364:
3363:
3358:
3353:
3348:
3343:
3338:
3332:
3330:
3324:
3323:
3321:
3320:
3315:
3307:
3299:
3291:
3285:
3283:
3279:
3278:
3276:
3275:
3270:
3265:
3260:
3255:
3250:
3245:
3239:
3237:
3233:
3232:
3230:
3229:
3227:Operator space
3224:
3219:
3214:
3209:
3204:
3199:
3194:
3189:
3187:Banach algebra
3184:
3178:
3176:
3175:Basic concepts
3172:
3171:
3160:
3159:
3152:
3145:
3137:
3129:
3128:
3124:John B. Conway
3116:
3086:
3077:
3045:
3008:
2990:
2967:
2965:, p. 317.
2955:
2943:Linear algebra
2928:
2927:
2925:
2922:
2919:
2918:
2900:
2899:
2897:
2894:
2893:
2892:
2886:
2879:
2876:
2875:
2874:
2869:
2864:
2859:
2854:
2843:
2842:Generalization
2840:
2836:
2835:
2824:
2821:
2816:
2812:
2808:
2803:
2798:
2795:
2792:
2789:
2784:
2768:
2767:
2755:
2752:
2747:
2743:
2739:
2736:
2733:
2730:
2727:
2724:
2691:
2690:
2679:
2674:
2670:
2666:
2663:
2660:
2655:
2651:
2631:
2628:
2598:
2595:
2573:
2569:
2545:
2544:
2529:
2526:
2523:
2521:
2519:
2516:
2513:
2510:
2505:
2501:
2497:
2492:
2488:
2484:
2481:
2476:
2472:
2468:
2465:
2462:
2459:
2456:
2454:
2452:
2449:
2446:
2441:
2437:
2431:
2427:
2423:
2420:
2417:
2414:
2411:
2406:
2402:
2398:
2393:
2389:
2385:
2382:
2379:
2376:
2373:
2371:
2369:
2366:
2358:
2350:
2347:
2342:
2338:
2332:
2328:
2324:
2321:
2318:
2315:
2312:
2309:
2304:
2300:
2296:
2291:
2287:
2283:
2280:
2277:
2274:
2271:
2269:
2267:
2264:
2259:
2255:
2249:
2245:
2241:
2236:
2232:
2228:
2225:
2222:
2219:
2216:
2211:
2207:
2203:
2198:
2194:
2190:
2187:
2184:
2181:
2178:
2176:
2174:
2171:
2166:
2162:
2156:
2152:
2148:
2143:
2138:
2134:
2130:
2127:
2124:
2121:
2118:
2113:
2109:
2105:
2100:
2095:
2091:
2087:
2084:
2081:
2078:
2075:
2073:
2071:
2068:
2063:
2059:
2053:
2049:
2043:
2039:
2035:
2030:
2026:
2022:
2019:
2016:
2013:
2010:
2005:
2001:
1995:
1991:
1987:
1982:
1978:
1974:
1971:
1968:
1965:
1962:
1960:
1958:
1954:
1948:
1944:
1938:
1934:
1928:
1924:
1920:
1915:
1911:
1907:
1902:
1898:
1892:
1888:
1884:
1879:
1875:
1870:
1866:
1863:
1860:
1857:
1855:
1853:
1848:
1844:
1840:
1837:
1834:
1831:
1828:
1827:
1809:
1808:
1793:
1788:
1784:
1778:
1774:
1768:
1764:
1760:
1755:
1751:
1747:
1742:
1738:
1732:
1728:
1724:
1719:
1715:
1711:
1708:
1706:
1704:
1699:
1695:
1689:
1685:
1681:
1676:
1672:
1668:
1663:
1658:
1653:
1650:
1645:
1641:
1637:
1634:
1632:
1630:
1625:
1621:
1615:
1611:
1605:
1601:
1595:
1591:
1587:
1582:
1577:
1572:
1569:
1564:
1560:
1556:
1553:
1551:
1547:
1543:
1539:
1536:
1535:
1495:
1487:
1478:
1464:
1456:
1447:
1440:
1436:
1429:
1422:
1415:
1407:
1391:
1383:
1367:
1331:
1328:
1309:
1308:
1294:
1289:
1285:
1281:
1278:
1275:
1270:
1265:
1261:
1240:
1235:
1231:
1227:
1224:
1221:
1216:
1212:
1191:
1169:
1165:
1142:
1138:
1101:
1096:
1092:
1069:
1066:
1044:
1024:
1004:
1001:
981:
959:
955:
934:
914:
892:
888:
867:
847:
836:
835:
824:
819:
815:
809:
805:
801:
796:
792:
786:
782:
761:
756:
750:
746:
742:
739:
733:
728:
724:
719:
697:
691:
687:
683:
680:
674:
669:
665:
642:
638:
634:
631:
626:
622:
618:
615:
606:is written as
595:
575:
564:
550:
546:
542:
537:
533:
529:
526:
523:
520:
517:
514:
509:
505:
501:
498:
495:
492:
489:
486:
483:
480:
475:
471:
467:
464:
459:
455:
451:
448:
445:
425:
405:
402:
397:
393:
389:
386:
383:
380:
377:
374:
364:
350:
346:
335:
323:
300:
266:
263:
251:
250:
228:
218:skew-Hermitian
215:
202:
117:
116:
31:
29:
22:
15:
9:
6:
4:
3:
2:
4532:
4521:
4518:
4516:
4513:
4512:
4510:
4495:
4487:
4486:
4483:
4477:
4474:
4472:
4469:
4467:
4466:Weak topology
4464:
4462:
4459:
4457:
4454:
4452:
4449:
4448:
4446:
4442:
4435:
4431:
4428:
4426:
4423:
4421:
4418:
4416:
4413:
4411:
4408:
4406:
4403:
4401:
4398:
4396:
4393:
4391:
4390:Index theorem
4388:
4386:
4383:
4381:
4378:
4376:
4373:
4372:
4370:
4366:
4360:
4357:
4355:
4352:
4351:
4349:
4347:Open problems
4345:
4339:
4336:
4334:
4331:
4329:
4326:
4324:
4321:
4319:
4316:
4314:
4311:
4310:
4308:
4304:
4298:
4295:
4293:
4290:
4288:
4285:
4283:
4280:
4278:
4275:
4273:
4270:
4268:
4265:
4263:
4260:
4258:
4255:
4253:
4250:
4249:
4247:
4243:
4237:
4234:
4232:
4229:
4227:
4224:
4222:
4219:
4217:
4214:
4212:
4209:
4207:
4204:
4202:
4199:
4197:
4194:
4193:
4191:
4189:
4185:
4175:
4172:
4170:
4167:
4165:
4162:
4159:
4155:
4151:
4148:
4146:
4143:
4141:
4138:
4137:
4135:
4131:
4125:
4122:
4120:
4117:
4115:
4112:
4110:
4107:
4105:
4102:
4100:
4097:
4095:
4092:
4090:
4087:
4085:
4082:
4080:
4077:
4076:
4073:
4070:
4066:
4061:
4057:
4053:
4046:
4041:
4039:
4034:
4032:
4027:
4026:
4023:
4011:
4010:
4005:
4003:
4001:
3997:
3993:
3989:
3985:
3984:
3982:
3978:
3972:
3969:
3967:
3964:
3962:
3959:
3957:
3954:
3952:
3949:
3947:
3944:
3942:
3939:
3937:
3934:
3932:
3929:
3928:
3926:
3922:
3915:
3911:
3908:
3906:
3903:
3901:
3898:
3897:
3895:
3893:Other results
3891:
3885:
3882:
3880:
3877:
3875:
3872:
3871:
3869:
3865:
3859:
3856:
3854:
3851:
3849:
3845:
3844:Hilbert space
3842:
3840:
3836:
3835:Inner product
3833:
3831:
3828:
3827:
3825:
3821:
3817:
3810:
3805:
3803:
3798:
3796:
3791:
3790:
3787:
3775:
3772:
3770:
3767:
3765:
3762:
3760:
3757:
3755:
3752:
3750:
3747:
3745:
3742:
3740:
3737:
3735:
3732:
3730:
3727:
3725:
3722:
3720:
3717:
3715:
3712:
3710:
3707:
3705:
3702:
3699:
3695:
3692:
3690:
3687:
3685:
3682:
3681:
3679:
3675:
3669:
3666:
3665:
3663:
3659:
3653:
3650:
3648:
3645:
3643:
3640:
3638:
3635:
3633:
3630:
3628:
3625:
3623:
3620:
3618:
3615:
3613:
3610:
3609:
3607:
3605:Miscellaneous
3603:
3596:
3592:
3589:
3587:
3584:
3582:
3579:
3577:
3574:
3573:
3571:
3567:
3561:
3558:
3556:
3553:
3551:
3548:
3546:
3543:
3542:
3540:
3536:
3528:
3525:
3524:
3523:
3520:
3518:
3515:
3513:
3510:
3508:
3505:
3503:
3500:
3498:
3494:
3492:
3489:
3488:
3486:
3482:
3476:
3473:
3471:
3468:
3466:
3463:
3461:
3458:
3456:
3453:
3451:
3448:
3446:
3443:
3441:
3438:
3436:
3433:
3432:
3430:
3426:
3420:
3417:
3415:
3412:
3410:
3407:
3405:
3402:
3400:
3397:
3393:
3390:
3388:
3385:
3383:
3380:
3379:
3378:
3375:
3374:
3372:
3370:Decomposition
3368:
3362:
3359:
3357:
3354:
3352:
3349:
3347:
3344:
3342:
3339:
3337:
3334:
3333:
3331:
3329:
3325:
3319:
3316:
3314:
3311:
3308:
3306:
3303:
3300:
3298:
3295:
3292:
3290:
3287:
3286:
3284:
3280:
3274:
3271:
3269:
3266:
3264:
3261:
3259:
3256:
3254:
3251:
3249:
3246:
3244:
3241:
3240:
3238:
3234:
3228:
3225:
3223:
3220:
3218:
3215:
3213:
3210:
3208:
3205:
3203:
3200:
3198:
3195:
3193:
3190:
3188:
3185:
3183:
3180:
3179:
3177:
3173:
3169:
3165:
3158:
3153:
3151:
3146:
3144:
3139:
3138:
3135:
3125:
3120:
3114:
3110:
3107:
3103:
3099:
3096:
3090:
3081:
3063:
3056:
3049:
3041:
3037:
3033:
3029:
3025:
3021:
3020:
3012:
2997:
2993:
2987:
2983:
2982:
2974:
2972:
2964:
2959:
2952:
2948:
2944:
2940:
2933:
2929:
2915:
2911:
2905:
2901:
2890:
2887:
2885:
2882:
2881:
2873:
2870:
2868:
2865:
2863:
2860:
2858:
2855:
2853:
2850:
2849:
2848:
2839:
2822:
2814:
2810:
2796:
2790:
2773:
2772:
2771:
2750:
2745:
2741:
2734:
2728:
2725:
2715:
2714:
2713:
2710:
2708:
2704:
2700:
2696:
2677:
2672:
2668:
2664:
2661:
2658:
2653:
2649:
2641:
2640:
2639:
2637:
2627:
2625:
2620:
2617:
2615:
2611:
2607:
2602:
2594:
2592:
2587:
2571:
2567:
2558:
2554:
2550:
2527:
2524:
2522:
2508:
2503:
2499:
2495:
2490:
2486:
2482:
2474:
2470:
2463:
2460:
2457:
2455:
2444:
2439:
2435:
2429:
2425:
2418:
2415:
2412:
2404:
2400:
2396:
2391:
2387:
2380:
2377:
2374:
2372:
2364:
2356:
2340:
2336:
2330:
2326:
2322:
2316:
2313:
2310:
2302:
2298:
2294:
2289:
2285:
2278:
2275:
2272:
2270:
2257:
2253:
2247:
2243:
2239:
2234:
2230:
2223:
2220:
2217:
2209:
2205:
2201:
2196:
2192:
2185:
2182:
2179:
2177:
2164:
2160:
2154:
2150:
2146:
2141:
2136:
2132:
2125:
2122:
2119:
2111:
2107:
2103:
2098:
2093:
2089:
2082:
2079:
2076:
2074:
2061:
2057:
2051:
2047:
2041:
2037:
2033:
2028:
2024:
2017:
2014:
2011:
2003:
1999:
1993:
1989:
1985:
1980:
1976:
1969:
1966:
1963:
1961:
1952:
1946:
1942:
1936:
1932:
1926:
1922:
1918:
1913:
1909:
1905:
1900:
1896:
1890:
1886:
1882:
1877:
1873:
1868:
1864:
1861:
1858:
1856:
1846:
1842:
1838:
1832:
1829:
1818:
1817:
1816:
1814:
1791:
1786:
1782:
1776:
1772:
1766:
1762:
1758:
1753:
1749:
1745:
1740:
1736:
1730:
1726:
1722:
1717:
1713:
1709:
1707:
1697:
1693:
1687:
1683:
1674:
1670:
1666:
1661:
1648:
1643:
1639:
1635:
1633:
1623:
1619:
1613:
1609:
1603:
1593:
1589:
1585:
1580:
1567:
1562:
1558:
1554:
1552:
1545:
1541:
1537:
1526:
1525:
1524:
1522:
1518:
1514:
1513:inner product
1510:
1506:
1502:
1498:
1490:
1486:
1482:
1474:
1469:
1467:
1459:
1455:
1451:
1443:
1432:
1425:
1418:
1410:
1406:
1402:
1398:
1394:
1386:
1382:
1378:
1374:
1370:
1363:
1359:
1357:
1353:
1349:
1345:
1341:
1337:
1327:
1325:
1320:
1318:
1314:
1292:
1287:
1283:
1279:
1276:
1273:
1268:
1263:
1259:
1238:
1233:
1229:
1225:
1222:
1219:
1214:
1210:
1189:
1167:
1163:
1140:
1136:
1127:
1126:
1125:
1123:
1118:
1116:
1099:
1094:
1090:
1064:
1042:
1022:
1002:
999:
979:
957:
953:
932:
912:
890:
886:
865:
845:
822:
817:
813:
807:
803:
799:
794:
790:
784:
780:
759:
754:
748:
744:
740:
737:
731:
726:
722:
717:
695:
689:
685:
681:
678:
672:
667:
663:
640:
636:
632:
629:
624:
620:
616:
613:
593:
573:
565:
548:
540:
535:
531:
524:
518:
515:
512:
507:
503:
499:
493:
487:
484:
481:
478:
473:
469:
462:
457:
449:
446:
423:
400:
395:
391:
384:
378:
375:
365:
348:
344:
336:
321:
314:
313:
312:
298:
289:
287:
283:
280:
276:
272:
262:
260:
256:
255:normal matrix
248:
244:
240:
236:
232:
229:
227:
223:
219:
216:
214:
210:
206:
203:
201:
197:
193:
190:
189:
188:
186:
181:
179:
175:
171:
168:
164:
160:
156:
152:
149:
146:
142:
139:
138:Hilbert space
136:
132:
128:
125:, especially
124:
113:
110:
102:
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: â
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
4456:Balanced set
4430:Distribution
4368:Applications
4271:
4221:KreinâMilman
4206:Closed graph
4008:
3999:
3995:
3991:
3987:
3956:Self-adjoint
3950:
3867:Main results
3677:Applications
3507:Disk algebra
3361:Spectral gap
3296:
3236:Main results
3119:
3089:
3080:
3069:. Retrieved
3048:
3023:
3017:
3011:
3000:. Retrieved
2980:
2958:
2942:
2932:
2904:
2845:
2837:
2769:
2711:
2706:
2702:
2698:
2694:
2692:
2635:
2633:
2621:
2618:
2613:
2609:
2605:
2603:
2600:
2588:
2552:
2546:
1810:
1520:
1516:
1508:
1504:
1500:
1493:
1488:
1484:
1476:
1472:
1470:
1462:
1457:
1453:
1445:
1434:
1427:
1420:
1413:
1408:
1404:
1400:
1396:
1389:
1384:
1380:
1376:
1372:
1365:
1361:
1360:
1355:
1351:
1347:
1343:
1339:
1335:
1333:
1321:
1310:
1119:
837:
290:
268:
258:
252:
246:
242:
238:
234:
225:
221:
212:
208:
199:
195:
182:
177:
173:
169:
158:
154:
150:
140:
130:
120:
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
4385:Heat kernel
4375:Hardy space
4282:Trace class
4196:HahnâBanach
4158:Topological
3966:Trace class
3704:Heat kernel
3404:Compression
3289:Isospectral
3026:: 337â340.
2604:An element
1399:stabilizes
220:operators:
172:, that is:
123:mathematics
4509:Categories
4318:C*-algebra
4133:Properties
3382:Continuous
3197:C*-algebra
3192:B*-algebra
3106:Uniqueness
3071:2011-07-01
3002:2021-06-26
2939:Kunze, Ray
2924:References
2912:of, e.g.,
2857:Normaloids
2624:C*-algebra
1499:). Since (
363:is normal.
334:is normal.
265:Properties
145:continuous
69:newspapers
4292:Unbounded
4287:Transpose
4245:Operators
4174:Separable
4169:Reflexive
4154:Algebraic
4140:Barrelled
3168:-algebras
3095:Existence
3040:124945750
2815:∗
2754:‖
2746:∗
2738:‖
2732:‖
2723:‖
2673:∗
2654:∗
2568:ℓ
2504:∗
2496:−
2491:∗
2464:
2440:∗
2419:
2413:−
2405:∗
2381:
2341:∗
2317:
2311:−
2303:∗
2279:
2258:∗
2224:
2218:−
2210:∗
2186:
2165:∗
2126:
2120:−
2112:∗
2083:
2052:∗
2018:
2012:−
1994:∗
1970:
1937:∗
1906:−
1891:∗
1865:
1847:∗
1833:
1777:∗
1746:−
1731:∗
1688:∗
1667:−
1614:∗
1586:−
1546:∗
1293:∗
1269:∗
1095:∗
1068:¯
1065:λ
1023:λ
891:∗
749:∗
741:−
690:∗
545:‖
536:∗
528:‖
522:⟩
508:∗
497:⟨
491:⟩
474:∗
466:⟨
454:‖
444:‖
404:‖
396:∗
388:‖
382:‖
373:‖
349:⋆
241:for some
165:with its
99:June 2011
4494:Category
4306:Algebras
4188:Theorems
4145:Complete
4114:Schwartz
4060:glossary
3980:Examples
3769:Weyl law
3714:Lax pair
3661:Examples
3495:With an
3414:Discrete
3392:Residual
3328:Spectrum
3313:operator
3305:operator
3297:operator
3212:Spectrum
3109:Archived
3098:Archived
3062:Archived
2996:Archived
2941:(1971),
2878:See also
1511:) is an
1426:= 0, or
992:for any
416:for all
163:commutes
153: :
4297:Unitary
4277:Nuclear
4262:Compact
4257:Bounded
4252:Adjoint
4226:Minâmax
4119:Sobolev
4104:Nuclear
4094:Hilbert
4089:Fréchet
4054: (
3994:) with
3971:Unitary
3830:Adjoint
3310:Unitary
2951:0276251
1507:) ⊠tr(
1468:) = 0.
135:complex
83:scholar
4272:Normal
4109:Orlicz
4099:Hölder
4079:Banach
4068:Spaces
4056:topics
3951:Normal
3294:Normal
3038:
2988:
2949:
1362:Proof.
85:
78:
71:
64:
56:
4084:Besov
4002:<â
3387:Point
3065:(PDF)
3058:(PDF)
3036:S2CID
2896:Notes
2770:with
1813:trace
1338:on a
1251:then
772:then
655:with
436:(use
284:) is
161:that
143:is a
133:on a
90:JSTOR
76:books
4432:(or
4150:Dual
3924:Maps
3846:and
3837:and
3318:Unit
3166:and
2986:ISBN
1471:Let
1364:Let
1315:and
1155:and
878:and
710:and
291:Let
273:. A
245:(so
129:, a
62:news
3028:doi
2707:NN*
2703:N*N
2614:x*x
2610:xx*
2553:AB*
1521:XX*
1509:AB*
1379:is
1128:If
838:If
239:MM*
224:= â
178:N*N
174:NN*
121:In
45:by
4511::
4058:â
3104:,
3060:.
3034:.
3024:14
3022:.
2994:.
2970:^
2947:MR
2695:N*
2626:.
2616:.
2612:=
2593:.
2528:0.
2461:tr
2416:tr
2378:tr
2314:tr
2276:tr
2221:tr
2183:tr
2123:tr
2080:tr
2015:tr
1967:tr
1862:tr
1830:tr
1503:,
1475:=
1439:TP
1433:=
1428:TP
1421:TP
1326:.
1319:.
732::=
673::=
563:).
288:.
261:.
253:A
237:=
233::
222:N*
211:=
209:N*
198:=
196:N*
194::
180:.
176:=
170:N*
157:â
4436:)
4160:)
4156:/
4152:(
4062:)
4044:e
4037:t
4030:v
4009:F
4000:n
3996:K
3992:K
3990:(
3988:C
3916:)
3912:(
3808:e
3801:t
3794:v
3700:)
3696:(
3597:)
3593:(
3156:e
3149:t
3142:v
3074:.
3042:.
3030::
3005:.
2823:.
2820:)
2811:N
2807:(
2802:D
2797:=
2794:)
2791:N
2788:(
2783:D
2751:x
2742:N
2735:=
2729:x
2726:N
2699:N
2678:.
2669:N
2665:N
2662:=
2659:N
2650:N
2636:N
2606:x
2572:2
2525:=
2515:)
2512:)
2509:T
2500:T
2487:T
2483:T
2480:(
2475:V
2471:P
2467:(
2458:=
2448:)
2445:T
2436:T
2430:V
2426:P
2422:(
2410:)
2401:T
2397:T
2392:V
2388:P
2384:(
2375:=
2365:V
2357:T
2346:)
2337:T
2331:V
2327:P
2323:T
2320:(
2308:)
2299:T
2295:T
2290:V
2286:P
2282:(
2273:=
2263:)
2254:T
2248:V
2244:P
2240:T
2235:V
2231:P
2227:(
2215:)
2206:T
2202:T
2197:V
2193:P
2189:(
2180:=
2170:)
2161:T
2155:V
2151:P
2147:T
2142:2
2137:V
2133:P
2129:(
2117:)
2108:T
2104:T
2099:2
2094:V
2090:P
2086:(
2077:=
2067:)
2062:V
2058:P
2048:T
2042:V
2038:P
2034:T
2029:V
2025:P
2021:(
2009:)
2004:V
2000:P
1990:T
1986:T
1981:V
1977:P
1973:(
1964:=
1953:)
1947:V
1943:P
1933:T
1927:V
1923:P
1919:T
1914:V
1910:P
1901:V
1897:P
1887:T
1883:T
1878:V
1874:P
1869:(
1859:=
1852:)
1843:X
1839:X
1836:(
1792:.
1787:V
1783:P
1773:T
1767:V
1763:P
1759:T
1754:V
1750:P
1741:V
1737:P
1727:T
1723:T
1718:V
1714:P
1710:=
1698:V
1694:P
1684:T
1680:)
1675:V
1671:P
1662:H
1657:1
1652:(
1649:T
1644:V
1640:P
1636:=
1624:V
1620:P
1610:T
1604:2
1600:)
1594:V
1590:P
1581:H
1576:1
1571:(
1568:T
1563:V
1559:P
1555:=
1542:X
1538:X
1517:H
1505:B
1501:A
1496:V
1494:P
1492:â
1489:H
1485:1
1483:(
1481:T
1479:V
1477:P
1473:X
1465:V
1463:P
1461:â
1458:H
1454:1
1452:(
1450:T
1448:V
1446:P
1441:V
1437:V
1435:P
1430:V
1423:V
1419:)
1416:V
1414:P
1412:â
1409:H
1405:1
1401:V
1397:T
1392:V
1390:P
1388:â
1385:H
1381:1
1377:V
1373:V
1368:V
1366:P
1356:T
1352:V
1348:V
1344:H
1336:T
1307:.
1288:2
1284:N
1280:A
1277:=
1274:A
1264:1
1260:N
1239:,
1234:2
1230:N
1226:A
1223:=
1220:A
1215:1
1211:N
1190:A
1168:2
1164:N
1141:1
1137:N
1100:.
1091:N
1043:N
1003:.
1000:k
980:N
958:k
954:N
933:N
913:N
887:N
866:N
846:N
823:.
818:1
814:T
808:2
804:T
800:=
795:2
791:T
785:1
781:T
760:,
755:2
745:T
738:T
727:2
723:T
718:i
696:2
686:T
682:+
679:T
668:1
664:T
641:2
637:T
633:i
630:+
625:1
621:T
617:=
614:T
594:T
574:T
549:2
541:x
532:T
525:=
519:x
516:,
513:x
504:T
500:T
494:=
488:x
485:,
482:x
479:T
470:T
463:=
458:2
450:x
447:T
424:x
401:x
392:T
385:=
379:x
376:T
345:T
322:T
299:T
259:C
247:N
243:M
235:N
226:N
213:N
200:N
159:H
155:H
151:N
141:H
112:)
106:(
101:)
97:(
87:·
80:·
73:·
66:·
39:.
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