Fuglede's theorem: Difference between revisions

→‎Putnam's generalization: explained where normality is used.
(Added {{Template:Functional analysis}})
(→‎Putnam's generalization: explained where normality is used.)
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> because <math>M</math> is normal, and similarly <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