Please check my proof (generalized eigenspaces)

[bennyska]

Homework Statement

let T be a linear operator on V, and let λ be an eigenvalue of T. prove that if rank((T-λI)^m = rank((T-λI)^m+1 for some integer m, then K_λ = N((T-λI)^m.

Homework Equations

this is part 4 of a 6 part proof, and in the earlier stages, i showed if rank((T-λI)^m = rank((T-λI)^m+1 for some integer m, then N((T-λI)^m = N((T-λI)^m+1.

The Attempt at a Solution

let x be in K_λ. by previous result, N((T-λI)^m = N((T-λI)^k for k≥ m. x is in K_λ, so x is in N((T-λI)^p for some integer p. if p≥m, then x is in N((T-λI)^m. if p<m, then m = p+t for some t. then (T-λI)^p+t(x) = (T-λI)^t(T-λI)^p(x) = (T-λI)^t(0)=0, so x is in N((T-λI)^m.
let y be in N((T-λI)^m. then y is in N((T-λI)^p for some integer p, so y in in K_λ, and the two sets are equal.

 

[bennyska]

well, i see a mistake already, in the "if p≥m" part...

 

[bennyska]

can i just say, since (T-λI)^p(x)=0, then x is N((T-λI)^m if m=p?