Chemistry - Fuglede's theorem

In mathematics, Fuglede's theorem is a result in functional analysis. The following version extends the original theorem.

Theorem (Fuglede - Putnam - Rosenblum): Let T, M, N be linear operators on a complex Banach space, and suppose that M and N are normal and MT = TN. Then M*T = TN*.

Proof: By induction, the hypothesis implies that M^kT = TN^k for all k. Thus for any λ in $\mathbb{C}$ ,

$e^{\bar\lambda M}T = T e^{\bar\lambda N}$ .

Consider the function

$F(\lambda) = e^{\lambda M^*} T e^{-\lambda N^*}$

This is equal to

$e^{\lambda M^*} \left[e^{-\bar\lambda M}T e^{\bar\lambda N}\right] e^{-\lambda N^*} = U(\lambda) T V(\lambda)^{-1}$ ,

where $U(\lambda) = e^{\lambda M^* - \bar\lambda M}$ and $V(\lambda) = e^{\lambda N^* - \bar\lambda N}$ . However we have

$U(\lambda)^* = e^{\bar\lambda M - \lambda M^*} = U(\lambda)^{-1}$

so U is unitary, and hence has norm 1 for all λ; the same is true for V(λ), so

$\|F(\lambda)\| \le \|T\|\ \forall \lambda$

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*.

History: The original paper of Fuglede dealt with the case M = N only, and 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.

Categories: Functional analysis | Theorems