Endomorphism/K/Powers/Boundedness/Fact

Let ${}V$ be a finite-dimensional ${}{\mathbb {K} }$-vector space, and let

$\varphi \colon V\longrightarrow V$

be an endomorphism. Then the following properties are equivalent.

1. ${}\varphi$ is stable.
2. For every ${}v\in V$, the sequence ${}\varphi ^{n}(v)$, ${}n\in \mathbb {N}$, is bounded.
3. There exists a generating system ${}v_{1},\ldots ,v_{m}\in V$ such that ${}\varphi ^{n}(v_{j})$, ${}j=1,\ldots ,m$, is bounded.
4. The modulus of every complex eigenvalue of ${}\varphi$ is smaller or equal ${}1$, and the eigenvalues of modulus ${}1$ are diagonalizable, that is, their algebraic multiplicity equals their geometric multiplicity.
5. For a describing matrix ${}M$ of ${}\varphi$, considered over ${}\mathbb {C}$, the Jordan blocks of the Jordan normal form have the form

   ${\begin{pmatrix}\lambda &1&0&\cdots &0\\0&\lambda &1&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\\vdots &\cdots &0&\lambda &1\\0&\cdots &\cdots &0&\lambda \end{pmatrix}}$

   with ${}\vert {\lambda }\vert <1$, or the form ${}(\lambda )$ with ${}\vert {\lambda }\vert =1$.

Proof, Write another proof