(a) Let $T_n:\mathbb{R}^n\rightarrow \mathbb{R}^n$ be defined by $T_n x=\left(0,\frac{\xi_1}{1},\frac{\xi_2}{2},\dots,\frac{\xi_{n-1}}{n-1}\right),$ where $x=(\xi_1,\dots,\xi_n).$ Find all eigenvalues and eigenvectors of $T_n$ and their algebraic and geometric multiplicities.
(b) Let $T:\ell^2\to\ell^2$ be defined by $Tx=\left(0,\frac{\xi_1}{1},\frac{\xi_2}{2},\frac{\xi_3}{3},\dots\right),$ where $x=(\xi_1,\xi_2,\dots).$
Show that $T$ has no eigenvalues, and that $\lambda=0$ belongs to $\sigma_r(T)$.
I don't know how to proceed. Does anyone have a similar example, or could point me in the right direction?
Best Answer
Write out (in an equation) what it means for $x$ to be an eigenvalue of $T$ in both cases. In part (b), you should see that this forces $x=0$ and in part (a) you should be able to classify all such $x$.
For example, in (a), if $Tx=\lambda x$ with $\lambda\neq 0$, then you must have $x_1=0$ which implies $x_2=0$, and so on. So we must have $\lambda=0$. Now can you find any $x$ that satisfies $Tx=0$?