STTS show that the generalized eigenspace of T are invariant

ST=TS show that the generalized eigenspace of T are invariant under T

Solution

Using that T commutes with both itself and lemda kI, we see that T commutes
T - lemda kI and hence with (T - lemda kI)n. Let v 2 Uk. This means (T - lemda kI)nv = 0.
Therefore
(T- lemda kI)nTv = T(T - lemda kI)nv = T(0) = 0
which implies Tv 2 Uk. Therefore Uk is T invariant.