Fuglede's theorem

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

Contenuti 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 _{io}lambda _{io}P_{io}} 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 (per esempio. 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 {stile di visualizzazione N^{*}=somma _{io}{{sbarra {lambda }}_{io}}P_{io}.} In generale, 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. così, it is not so obvious that T also commutes with any simple function of the form {displaystyle rho =sum _{io}{sbarra {lambda }}P_{Omega _{io}}.} Infatti, 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 {stile di visualizzazione P_{Omega _{io}}} , 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.

Teorema (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*.

Prima prova (Marvin Rosenblum): Per induzione, the hypothesis implies that MkT = TNk for all k. Thus for any λ in {displaystyle mathbb {C} } , {stile di visualizzazione e^{{sbarra {lambda }}M}T=Te^{{sbarra {lambda }}N}.} Consider the function {stile di visualizzazione F(lambda )=e^{lambda M^{*}}Te^{-lambda N^{*}}.} Questo è uguale a {stile di visualizzazione e^{lambda M^{*}}sinistra[e^{-{sbarra {lambda }}M}Te^{{sbarra {lambda }}N}Giusto]e^{-lambda N^{*}}=U(lambda )TV(lambda )^{-1},} dove {stile di visualizzazione U(lambda )=e^{lambda M^{*}-{sbarra {lambda }}M}} perché {stile di visualizzazione M} is normal, e similmente {stile di visualizzazione V(lambda )=e^{lambda N^{*}-{sbarra {lambda }}N}} . However we have {stile di visualizzazione U(lambda )^{*}=e^{{sbarra {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), Così {stile di visualizzazione |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: Seconda prova: Consider the matrices {displaystyle T'={inizio{bmatrice}0&0\T&0end{bmatrice}}quad {testo{e}}quad N'={inizio{bmatrice}N&0\0&Mend{bmatrice}}.} The operator N' is normal and, per assunzione, T' N' = N' T' . By Fuglede's theorem, uno ha {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, allora sono unitariamente equivalenti.

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

{stile di visualizzazione S^{-1}M^{*}S=N^{*}.} Take the adjoint of the above equation and we have {stile di visualizzazione S^{*}M(S^{-1})^{*}=N.} Così {stile di visualizzazione 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. Quindi {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.

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

Teorema (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*.

Riferimenti ^ Salta su: a b Putnam, C. R. (aprile 1951). "On Normal Operators in Hilbert Space". Giornale americano di matematica. 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, Testi di laurea in Matematica, vol. 15, New York-Heidelberg-Berlin: Springer-Verlag, p. 274, ISBN 0-387-90080-2, SIG 0417727. Rudino, Walter (1973). Analisi funzionale. Serie internazionale di matematica pura e applicata. vol. 25 (First ed.). New York, New York: McGraw-Hill Scienze/Ingegneria/Matematica. ISBN 9780070542259. nascondi vte Analisi funzionale (argomenti – glossario) Spazi BanachBesovFréchetHilbertHölderNucleareOrliczSchwartzSobolevVettore topologico Proprietà barrelledcompletatodual (algebrico/topologico)localmente convessoriflessivoseparabile TeoremiHahn–BanachRieszrappresentazionegrafo chiusoprincipio di limitatezza uniformeKakutani punto fissoKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Operatori adjointboundedcompactHilbert–Schmidtnormalnucleartrace classtransposeunboundedunitary Algebres Algebra di BanachC*-algebraspettro di un'algebra C*problemi di un operatore algebra localmente compatto di un'algebra di Neumanngruppo compatto di un'algebra di Neumann Problema del sottospazio Congettura di Mahler Applicazioni Spazio di Hardy Teoria spettrale delle equazioni differenziali ordinarie Heat Kernel Teorema dell'indice Calcolo delle variazioni Calcolo funzionale Operatore integrale Polinomio di Jones Teoria dei campi quantistici topologici Geometria non commutativa Ipotesi di Riemann Distribuzione (o funzioni generalizzate) Argomenti avanzati proprietà di approssimazione insieme bilanciato Teoria di Choquet topologia debole Distanza di Banach–Mazur Teoria di Tomita–Takesaki Categorie: Teoria degli operatori Teoremi nell'analisi funzionale