$\mathbf {The \ Problems \ are}:$
$\mathbf {1)}$ Let, $V$ be a finite dimensional vector space of dimension $n$ over an algebraically closed field $\mathbb F$ where $n \gt 2$. Does there exists a linear mapping $ \operatorname T : V \to V$ such that there are exactly $k(\lt n)$ invariant subspaces of $V$ under $\operatorname T$, where $k \in \mathbb N$ ???
$\mathbf {2)}$ Let, $\operatorname T : V \to V$ be a given transformation on $V$ for $\operatorname{dim} V \lt \infty$ and let the characteristic polynomial of the corresponding matrix of transformation $A$ be $$\prod_{j=1} ^ k (\lambda-\lambda_j)^{n_j}$$ where all $\lambda_j \in \mathbb F$ and if dimension of eigenspace of each $\lambda_j \ is \ 1$, then find the number of invariant subspaces of $V \ under \ \operatorname T .$
$\mathbf {My \ approach} :$ As $\operatorname{dim} V = n$, then let an ordered bases of $V$ be $\beta = \{\beta_1, \beta_2, \cdots ,\beta_n\}$ . Now, total number of invariant subspaces of $V$ are obviously $\leq 2^n-1$ and one can always find $2^p-1$ invariant subspaces for $p \lt n$ just by having $\operatorname T: V \to V$ such that $\operatorname T(\beta_i) = \beta_i \ \forall i \in \{1,2,\cdots,p\}
\ and \ \operatorname T(\beta_i) = 0 \ \forall i \in \{p+1,p+2,\cdots,n\}.$
But, the stated $k$ in the question may not be always of the form $2^p-1$ for some $p$ ???
For the 2nd problem, I am completely unable to approach , any hint is appreciated .
Best Answer
Note that $T$ admits a finite number of invariant subspaces IFF $T$ is cyclic, that is, $T$ satisfies the condition in 2) (when the $(\lambda_j)$ are distinct).
More precisely, if $T$ is cyclic, then, up to a change of basis, we may assume that $T=diag(\lambda_1I_{i_1}+J_1,\cdots,\lambda_kI_{i_k}+J_k)$ where $J_j$ is a nilpotent Jordan matrix and the $(\lambda_j)$ are distinct. Then $U$ is $T$-invariant IFF $U$ is the direct sum of $\lambda_jI_{i_j}+J_j$-invariant subspaces. Thus, it suffices to find the $J$-invariant subspaces, for some $n\times n$ nilpotent Jordan block $J$. We find the $n+1$ subspaces $\{0\},[e_1],[e_1,e_2],\cdots,F^n$.
Finally, the solution of 2) is $(n_1+1)\cdots(n_k+1)$ and the solution of 1) is NO (if $k\leq n)$ because $(n_1+1)\cdots(n_k+1)\geq n_1+\cdots+n_k+1$.