Fuglede's theorem

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

Contents 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 _{i}lambda _{i}P_{i}} 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 (e.g. 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 {displaystyle N^{*}=sum _{i}{{bar {lambda }}_{i}}P_{i}.} In general, when the Hilbert space is not finite-dimensional, the normal operator N gives rise to a projection-valued measure P on its spectrum, σ(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. Thus, it is not so obvious that T also commutes with any simple function of the form {displaystyle rho =sum _{i}{bar {lambda }}P_{Omega _{i}}.} Indeed, 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 {displaystyle P_{Omega _{i}}} , 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.

Theorem (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*.

First proof (Marvin Rosenblum): By induction, the hypothesis implies that MkT = TNk for all k. Thus for any λ in {displaystyle mathbb {C} } , {displaystyle e^{{bar {lambda }}M}T=Te^{{bar {lambda }}N}.} Consider the function {displaystyle F(lambda )=e^{lambda M^{*}}Te^{-lambda N^{*}}.} This is equal to {displaystyle e^{lambda M^{*}}left[e^{-{bar {lambda }}M}Te^{{bar {lambda }}N}right]e^{-lambda N^{*}}=U(lambda )TV(lambda )^{-1},} where {displaystyle U(lambda )=e^{lambda M^{*}-{bar {lambda }}M}} because {displaystyle M} is normal, and similarly {displaystyle V(lambda )=e^{lambda N^{*}-{bar {lambda }}N}} . However we have {displaystyle 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), so {displaystyle |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: Second proof: Consider the matrices {displaystyle T'={begin{bmatrix}0&0\T&0end{bmatrix}}quad {text{and}}quad N'={begin{bmatrix}N&0\0&Mend{bmatrix}}.} The operator N' is normal and, by assumption, T' N' = N' T' . By Fuglede's theorem, one has {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, then they are unitarily equivalent.

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

{displaystyle S^{-1}M^{*}S=N^{*}.} Take the adjoint of the above equation and we have {displaystyle S^{*}M(S^{-1})^{*}=N.} So {displaystyle 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. Then {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.

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

Theorem (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*.

References ^ Jump up to: a b Putnam, C. R. (April 1951). "On Normal Operators in Hilbert Space". American Journal of Mathematics. 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, Graduate Texts in Mathematics, vol. 15, New York-Heidelberg-Berlin: Springer-Verlag, p. 274, ISBN 0-387-90080-2, MR 0417727. Rudin, Walter (1973). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 25 (First ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 9780070542259. hide vte Functional analysis (topics – glossary) Spaces BanachBesovFréchetHilbertHölderNuclearOrliczSchwartzSobolevtopological vector Properties barrelledcompletedual (algebraic/topological)locally convexreflexiveseparable Theorems Hahn–BanachRiesz representationclosed graphuniform boundedness principleKakutani fixed-pointKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Operators adjointboundedcompactHilbert–Schmidtnormalnucleartrace classtransposeunboundedunitary Algebras Banach algebraC*-algebraspectrum of a C*-algebraoperator algebragroup algebra of a locally compact groupvon Neumann algebra Open problems invariant subspace problemMahler's conjecture Applications Hardy spacespectral theory of ordinary differential equationsheat kernelindex theoremcalculus of variationsfunctional calculusintegral operatorJones polynomialtopological quantum field theorynoncommutative geometryRiemann hypothesisdistribution (or generalized functions) Advanced topics approximation propertybalanced setChoquet theoryweak topologyBanach–Mazur distanceTomita–Takesaki theory Categories: Operator theoryTheorems in functional analysis

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