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.

${}\varphi$ is stable.
For every ${}v\in V$ , the sequence ${}\varphi ^{n}(v)$ , ${}n\in \mathbb {N}$ , is bounded.
There exists a generating system ${}v_{1},\ldots ,v_{m}\in V$ such that ${}\varphi ^{n}(v_{j})$ , ${}j=1,\ldots ,m$ , is bounded.
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.
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 ${}\vert {\lambda }\vert =1$ .