Eigenvalue kind of nilpotent problem

[jostpuur]

If a vector v\in V and a linear mapping T:V\to V are fixed, and there exists numbers \lambda_1\in\mathbb{C}, n_1\in\mathbb{N} so that

<br /> (T - \lambda_1)^{n_1}v = 0,<br />

is it possible that there exists some \lambda_2\neq\lambda_1, and n_2\in\mathbb{N} so that

<br /> (T - \lambda_2)^{n_2}v = 0?<br />

(Here complex numbers are interpreted as multiplication operators V\to V, v\mapsto \lambda v, as usual.)

 

[morphism]

That happens iff v=0. This is the statement that the intersection of different generalized eigenspaces is {0}.

 

[gel]

Furthermore, if f,g are any two polynomials with no common factor and f(T)v=g(T)v=0 then you can conclude that v=0.