Fuglede's theorem
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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 MkT = TNk for all k. Thus for any λ in <math>\mathbb{C}<math>,
- <math>e^{\bar\lambda M}T = T e^{\bar\lambda N}<math>.
Consider the function
- <math>F(\lambda) = e^{\lambda M^*} T e^{-\lambda N^*}<math>
This is equal to
- <math>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>U(\lambda) = e^{\lambda M^* - \bar\lambda M}<math> and <math>V(\lambda) = e^{\lambda N^* - \bar\lambda N}<math>. However we have
- <math>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>\|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*.
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.