Satz von Fuglede

Fuglede's theorem In mathematics, Fuglede's theorem is a result in operator theory, named after Bent Fuglede.

Inhalt 1 The result 2 Putnam's generalization 3 C*-algebras 4 References The result Theorem (Fuglede) Let T and N be bounded operators on a complex Hilbert space with N being normal. If TN = NT, then TN* = N*T, where N* denotes the adjoint of N.

Normality of N is necessary, as is seen by taking T=N. When T is self-adjoint, the claim is trivial regardless of whether N is normal: {displaystyle TN^{*}=(NT)^{*}=(TN)^{*}=N^{*}T.} Tentative Proof: If the underlying Hilbert space is finite-dimensional, the spectral theorem says that N is of the form {displaystyle N=sum _{ich}Lambda _{ich}P_{ich}} where Pi are pairwise orthogonal projections. One expects that TN = NT if and only if TPi = PiT. Indeed it can be proved to be true by elementary arguments (z.B. it can be shown that all Pi are representable as polynomials of N and for this reason, if T commutes with N, it has to commute with Pi...). Therefore T must also commute with {Anzeigestil N^{*}= Summe _{ich}{{Bar {Lambda }}_{ich}}P_{ich}.} Im Algemeinen, when the Hilbert space is not finite-dimensional, the normal operator N gives rise to a projection-valued measure P on its spectrum, p(N), which assigns a projection PΩ to each Borel subset of σ(N). N can be expressed as {displaystyle N=int _{Sigma (N)}lambda dP(Lambda ).} Differently from the finite dimensional case, it is by no means obvious that TN = NT implies TPΩ = PΩT. Daher, it is not so obvious that T also commutes with any simple function of the form {displaystyle rho =sum _{ich}{Bar {Lambda }}P_{Omega _{ich}}.} In der Tat, following the construction of the spectral decomposition for a bounded, normal, not self-adjoint, operator T, one sees that to verify that T commutes with {Anzeigestil P_{Omega _{ich}}} , the most straightforward way is to assume that T commutes with both N and N*, giving rise to a vicious circle!

That is the relevance of Fuglede's theorem: The latter hypothesis is not really necessary.

Putnam's generalization The following contains Fuglede's result as a special case. The proof by Rosenblum pictured below is just that presented by Fuglede for his theorem when assuming N=M.

Satz (Calvin Richard Putnam)[1] Let T, M, N be linear operators on a complex Hilbert space, and suppose that M and N are normal, T is bounded and MT = TN. Then M*T = TN*.

Erster Beweis (Marvin Rosenblum): Durch Induktion, the hypothesis implies that MkT = TNk for all k. Thus for any λ in {Anzeigestil mathbb {C} } , {Anzeigestil e^{{Bar {Lambda }}M}T=Te^{{Bar {Lambda }}N}.} Consider the function {Anzeigestil F(Lambda )=e^{lambda M^{*}}Te^{-lambda N^{*}}.} This is equal to {Anzeigestil e^{lambda M^{*}}links[e^{-{Bar {Lambda }}M}Te^{{Bar {Lambda }}N}Rechts]e^{-lambda N^{*}}=U(Lambda )TV(Lambda )^{-1},} wo {Anzeigestil U(Lambda )=e^{lambda M^{*}-{Bar {Lambda }}M}} Weil {Anzeigestil M} is normal, und ähnlich {Anzeigestil V(Lambda )=e^{lambda N^{*}-{Bar {Lambda }}N}} . However we have {Anzeigestil U(Lambda )^{*}=e^{{Bar {Lambda }}M-lambda M^{*}}=U(Lambda )^{-1}} so U is unitary, and hence has norm 1 for all λ; the same is true for V(l), Also {Anzeigestil |F(Lambda )|leq |T| forall lambda .} So F is a bounded analytic vector-valued function, and is thus constant, and equal to F(0) = T. Considering the first-order terms in the expansion for small λ, we must have M*T = TN*.

The original paper of Fuglede appeared in 1950; it was extended to the form given above by Putnam in 1951.[1] The short proof given above was first published by Rosenblum in 1958; it is very elegant, but is less general than the original proof which also considered the case of unbounded operators. Another simple proof of Putnam's theorem is as follows: Zweiter Beweis: Consider the matrices {displaystyle T'={Start{bMatrix}0&0\T&0end{bMatrix}}Quad {Text{und}}quad N'={Start{bMatrix}N&0\0&Mend{bMatrix}}.} The operator N' is normal and, nach Annahme, T' N' = N' T' . By Fuglede's theorem, hat man {displaystyle T'(N')^{*}=(N')^{*}T'.} Comparing entries then gives the desired result.

From Putnam's generalization, one can deduce the following: Corollary If two normal operators M and N are similar, dann sind sie einheitlich äquivalent.

Nachweisen: Suppose MS = SN where S is a bounded invertible operator. Putnam's result implies M*S = SN*, d.h.

{Anzeigestil S^{-1}M^{*}S=N^{*}.} Take the adjoint of the above equation and we have {Anzeigestil S^{*}M(S^{-1})^{*}=N.} So {Anzeigestil S^{*}M(S^{-1})^{*}=S^{-1}MSquad Rightarrow quad SS^{*}M(SS^{*})^{-1}=M.} Let S*=VR, with V a unitary (since S is invertible) and R the positive square root of SS*. As R is a limit of polynomials on SS*, the above implies that R commutes with M. It is also invertible. Dann {displaystyle N=S^{*}M(S^{*})^{-1}=VRMR^{-1}V^{*}=VMV^{*}.} Corollary If M and N are normal operators, and MN = NM, then MN is also normal.

Nachweisen: The argument invokes only Fuglede's theorem. One can directly compute {Anzeigestil (MN)(MN)^{*}=MN(NM)^{*}=MNM^{*}N^{*}.} By Fuglede, the above becomes {displaystyle =MM^{*}NN^{*}=M^{*}MN^{*}N.} But M and N are normal, Also {displaystyle =M^{*}N^{*}MN=(MN)^{*}MN.} C*-algebras The theorem can be rephrased as a statement about elements of C*-algebras.

Satz (Fuglede-Putnam-Rosenblum) Let x, y be two normal elements of a C*-algebra A and z such that xz = zy. Then it follows that x* z = z y*.

Referenzen ^ Hochspringen zu: a b Putnam, C. R. (April 1951). "On Normal Operators in Hilbert Space". Amerikanisches Journal für Mathematik. 73 (2): 357–362. doi:10.2307/2372180. Fuglede, Bent. A Commutativity Theorem for Normal Operators — PNAS Berberian, Sterling K. (1974), Lectures in Functional Analysis and Operator Theory, Abschlusstexte in Mathematik, vol. 15, New York-Heidelberg-Berlin: Springer-Verlag, p. 274, ISBN 0-387-90080-2, HERR 0417727. Rudin, Walter (1973). Funktionsanalyse. International Series in Pure and Applied Mathematics. Vol. 25 (First ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 9780070542259. verbergen vte Funktionsanalyse (Themen – Glossar) Leerzeichen BanachBesovFréchetHilbertHölderNuclearOrliczSchwartzSobolevtopological vector Properties barrelledcompletedual (algebraisch/topologisch)lokal konvexreflexivseparable Theoreme Hahn-BanachRiesz-Darstellunggeschlossener Graphgleichmäßiges BeschränktheitsprinzipKakutani-FixpunktKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Operatoren adjointboundcompactHilbert–Schmidtnormalnucleartrace classtransposeunboundedunitary Algebren Banach-AlgebraC*-AlgebraSpektrum einer C*-AlgebraOperator-Algebravon Gruppenalgebra einer lokalvariant-kompakten Gruppe SubraumproblemMahlersche Vermutung Anwendungen Hardy-RaumSpektraltheorie gewöhnlicher DifferentialgleichungenWärmekernindexsatzVariationsrechnungFunktionsrechnungIntegraloperatorJones-PolynomTopologische QuantenfeldtheorieNichtkommutative GeometrieRiemann-HypotheseVerteilung (oder verallgemeinerte Funktionen) Fortgeschrittene Themen Approximation PropertyBalanced SetChoquet-TheorieSchwache TopologieBanach-Mazur-AbstandTomita-Takesaki-Theorie Kategorien: OperatortheorieTheoreme in der Funktionalanalysis