Fuglede's theorem

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

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:

[math]\displaystyle{ TN^* = (NT)^* = (TN)^* = N^*T. }[/math]

Tentative Proof: If the underlying Hilbert space is finite-dimensional, the spectral theorem says that N is of the form [math]\displaystyle{ N = \sum_i \lambda_i P_i }[/math] where P_i are pairwise orthogonal projections. One expects that TN = NT if and only if TP_i = P_iT. Indeed, it can be proved to be true by elementary arguments (e.g. it can be shown that all P_i are representable as polynomials of N and for this reason, if T commutes with N, it has to commute with P_i...). Therefore T must also commute with [math]\displaystyle{ N^* = \sum_i {\bar \lambda_i} P_i. }[/math]

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 P_Ω to each Borel subset of σ(N). N can be expressed as [math]\displaystyle{ N = \int_{\sigma(N)} \lambda d P(\lambda). }[/math]

Differently from the finite dimensional case, it is by no means obvious that TN = NT implies TP_Ω = P_ΩT. Thus, it is not so obvious that T also commutes with any simple function of the form [math]\displaystyle{ \rho = \sum_i {\bar \lambda} P_{\Omega_i}. }[/math]

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 [math]\displaystyle{ P_{\Omega_i} }[/math], 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) 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 M^kT = TN^k for all k. Thus for any λ in [math]\displaystyle{ \Complex }[/math], [math]\displaystyle{ e^{\bar\lambda M}T = T e^{\bar\lambda N}. }[/math]

Consider the function [math]\displaystyle{ F(\lambda) = e^{\lambda M^*} T e^{-\lambda N^*}. }[/math] This is equal to [math]\displaystyle{ e^{\lambda M^*} \left[e^{-\bar\lambda M}T e^{\bar\lambda N}\right] e^{-\lambda N^*} = U(\lambda) T V(\lambda)^{-1}, }[/math] where [math]\displaystyle{ U(\lambda) = e^{\lambda M^* - \bar\lambda M} }[/math] because [math]\displaystyle{ M }[/math] is normal, and similarly [math]\displaystyle{ V(\lambda) = e^{\lambda N^* - \bar\lambda N} }[/math]. However we have [math]\displaystyle{ U(\lambda)^* = e^{\bar\lambda M - \lambda M^*} = U(\lambda)^{-1} }[/math] so U is unitary, and hence has norm 1 for all λ; the same is true for V(λ), so [math]\displaystyle{ \|F(\lambda)\| \le \|T\|\ \forall \lambda. }[/math]

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. 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

[math]\displaystyle{ T' = \begin{bmatrix} 0 & 0 \\ T & 0 \end{bmatrix} \quad \text{and} \quad N' = \begin{bmatrix} N & 0 \\ 0 & M \end{bmatrix}. }[/math]

The operator N' is normal and, by assumption, T' N' = N' T' . By Fuglede's theorem, one has [math]\displaystyle{ T' (N')^* = (N')^*T'. }[/math]

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. [math]\displaystyle{ S^{-1} M^* S = N^*. }[/math]

Take the adjoint of the above equation and we have [math]\displaystyle{ S^* M (S^{-1})^* = N. }[/math]

So [math]\displaystyle{ S^* M (S^{-1})^* = S^{-1} M S \quad \Rightarrow \quad SS^* M (SS^*)^{-1} = M. }[/math]

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 [math]\displaystyle{ N = S^*M (S^*)^{-1}=VRMR^{-1}V^*=VMV^*. }[/math]

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 [math]\displaystyle{ (MN) (MN)^* = MN (NM)^* = MN M^* N^*. }[/math]

By Fuglede, the above becomes [math]\displaystyle{ = M M^* N N^* = M^* M N^*N. }[/math]

But M and N are normal, so [math]\displaystyle{ = M^* N^* MN = (MN)^* MN. }[/math]

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

• Fuglede, Bent. A Commutativity Theorem for Normal Operators — PNAS
• Berberian, Sterling K. (1974), Lectures in Functional Analysis and Operator Theory, Graduate Texts in Mathematics, 15, New York-Heidelberg-Berlin: Springer-Verlag, p. 274, ISBN 0-387-90080-2 .
• Rudin, Walter (1973). Functional Analysis. International Series in Pure and Applied Mathematics. 25 (First ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 9780070542259. https://archive.org/details/functionalanalys00rudi.