On linear operators
Posted: Wed Jan 03, 2018 8:34 pm
Let $\alpha \in \mathbb{R} \setminus \{0\} $ and suppose that $F ,G$ are linear operators from $\mathbb{R}^n$ into $\mathbb{R}^n$ satisfying
\begin{equation*}F\circ G - G \circ F =\alpha F \end{equation*}
\begin{equation*}F\circ G - G \circ F =\alpha F \end{equation*}
- Show that for all k \in \mathbb{N} one has
\[F^k \circ G - G \circ F ^k= \alpha k F^k\] - Show that there exists k \geq 1 such that F^k =0
.