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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 MkT = TNk for all k.
Thus for any λ in 
,
.
Consider the function
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}](a/../upload/math/2e0e988ee1bd6acaf580b99444a1454c.png)
,
where 
and 
. However we have
so U is unitary, and hence has norm 1 for all λ; the same is true for V(λ), so
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.
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