teorema de Fuglede

Teorema de Fuglede em matemática, O teorema de Fuglede é um resultado da teoria dos operadores, named after Bent Fuglede.

Conteúdo 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 _{eu}lambda _{eu}P_{eu}} 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 (por exemplo. 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 {estilo de exibição N^{*}=soma _{eu}{{bar {lambda }}_{eu}}P_{eu}.} No geral, 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. Desta forma, it is not so obvious that T also commutes with any simple function of the form {displaystyle rho =sum _{eu}{bar {lambda }}P_{Ômega _{eu}}.} De fato, 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 {estilo de exibição P_{Ômega _{eu}}} , 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*.

Primeira prova (Marvin Rosenblum): Por indução, the hypothesis implies that MkT = TNk for all k. Thus for any λ in {estilo de exibição mathbb {C} } , {estilo de exibição e^{{bar {lambda }}M}T=Te^{{bar {lambda }}N}.} Consider the function {estilo de exibição F(lambda )=e^{lambda M^{*}}Te^{-lambda N^{*}}.} This is equal to {estilo de exibição e^{lambda M^{*}}deixei[e^{-{bar {lambda }}M}Te^{{bar {lambda }}N}certo]e^{-lambda N^{*}}=U(lambda )TV(lambda )^{-1},} Onde {estilo de exibição U(lambda )=e^{lambda M^{*}-{bar {lambda }}M}} Porque {estilo de exibição M} is normal, e da mesma forma {estilo de exibição V(lambda )=e^{lambda N^{*}-{bar {lambda }}N}} . However we have {estilo de exibição 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(λ), assim {estilo de exibição |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: Segunda prova: Consider the matrices {displaystyle T'={começar{bmatriz}0&0\T&0end{bmatriz}}quadrilátero {texto{e}}quad N'={começar{bmatriz}N&0\0&Mend{bmatriz}}.} The operator N' is normal and, por suposição, T' N' = N' T' . By Fuglede's theorem, um tem {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, então eles são unitariamente equivalentes.

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

{estilo de exibição S^{-1}M^{*}S=N^{*}.} Take the adjoint of the above equation and we have {estilo de exibição S^{*}M(S^{-1})^{*}=N.} Então {estilo de exibição 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. Então {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 {estilo de exibição (MN)(MN)^{*}=MN(NM)^{*}=MNM^{*}N^{*}.} By Fuglede, the above becomes {displaystyle =MM^{*}NN^{*}=M^{*}MN^{*}N.} But M and N are normal, assim {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*.

Referências ^ Ir para: a b Putnam, C. R. (abril 1951). "On Normal Operators in Hilbert Space". Revista Americana de Matemática. 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, Textos de Graduação em Matemática, volume. 15, New York-Heidelberg-Berlin: Springer-Verlag, p. 274, ISBN 0-387-90080-2, SENHOR 0417727. Rudin, Walter (1973). Análise funcional. International Series in Pure and Applied Mathematics. Volume. 25 (First ed.). Nova york, Nova Iorque: McGraw-Hill Science/Engineering/Math. ISBN 9780070542259. esconder vte análise funcional (tópicos – glossário) Espaços BanachBesovFréchetHilbertHölderNuclearOrliczSchwartzSobolevvetor topológico Propriedades barrilledcompletedual (algébrico/topológico)Teoremas localmente convexo, reflexivo e separável Representação de Hahn–BanachRiesz grafo fechadoprincípio da limitação uniformeKakutani ponto fixoKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Operadores adjointboundedcompactHilbert–Schmidtnormalnucleartrace classtransposeunboundedunitary ÁlgebrasBanach álgebraC*-algebraspectrum of a C*-algebraoperator álgebra álgebra de grupo de um grupo localmente compacto álgebra de von Neumann Problemas abertos invariantes problema de subespaço Conjectura de Mahler Aplicações Espaço de Hardy teoria espectral de equações diferenciais ordinárias núcleo de calor teorema de índice cálculo de variações cálculo funcional operador integral polinômio de Jones teoria quântica topológica de campos geometria não comutativa hipótese de Riemann distribuição (ou funções generalizadas) Tópicos avançados propriedade de aproximação conjunto equilibrado Teoria de Choquet topologia fraca Distância de Banach–Mazur Teoria de Tomita–Takesaki Categorias: Teoria dos operadoresTeoremas em análise funcional