Let T elementof LaplaceRopf3 be defined by Tx y z y x 0 Fin

Let T elementof Laplace(Ropf^3) be defined by T(x, y, z) = (-y, x, 0). Find all eigenvalues of T. Is T nilpotent?

Solution

(a)

T(x,y,z)=k(x,y,z)=(-y,x,0)

k=0 implies:x=y=z=0 ie trivial eigenvector and eigenvalue

So let k non zero so z must be 0

-y=kx

x=ky

-y=k^2y

y(k^2+1)=0

Eigenvalues are: k=i,-i

b)

T(x,y,z)=(-y,x.0)

T^2(x,y,z)=T(-y,x,0)=(-x,-y,0)=-(x,y,0)

...

T^{2n}=(-1)^n(x,y,0)

T^{2n+1}=(-1)^n(-y,x,0)

So T is not nilpotent