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On linear operators

Linear Algebra
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On linear operators


Post by Riemann » 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*}
  1. Show that for all k \in \mathbb{N} one has

    \[F^k \circ G - G \circ F ^k= \alpha k F^k\]
  2. Show that there exists k \geq 1 such that F^k =0
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$
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