Fuglede's theorem

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 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

$${\displaystyle N^{*}=\sum _{i}{{\bar {\lambda }}_{i}}P_{i}.}$$

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 P_Ω 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 TP_Ω = P_ΩT. 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.

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 M^kT = TN^k 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

$${\displaystyle T'={\begin{bmatrix}0&0\\T&0\end{bmatrix}}\quad {\text{and}}\quad N'={\begin{bmatrix}N&0\\0&M\end{bmatrix}}.}$$

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

$${\displaystyle S^{*}M(S^{-1})^{*}=S^{-1}MS\quad \Rightarrow \quad SS^{*}M(SS^{*})^{-1}=M.}$$

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

$${\displaystyle =M^{*}N^{*}MN=(MN)^{*}MN.}$$

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*.

• 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.