Let $T$ be a linear operator on a space $V$ of dimension $n$. Suppose there are $V_0,...,V_n$ of $V$ such that $\text{dim}(V_i)=i$ for all $i$ and $T(V_i)\subset V_{i-1}$ for all $i\geq 1$. What are the possible eigenvalues of $T$? When is $T$ diagonalizable?
My try: By definition, we know that $V_0=\{0\}$ and thus $T(V_1)=\{0\}$. Hence, $0$ is one of the eigenvalues of $T$. But how to find other eigenvalues?
Since $T(V_1)=0$, $T^{n+1}=0, T(x)=cx$ implies $T^{n+1}(x)=c^{n+1}x=0, c=0$.